xx}^{2}+\frac{1}{2}c_{6}^{4}c_{16}\chi_{1}\varepsilon^{\theta}\gamma^{-3}y_{\varepsilon}^{3}(t),$

because $\int_{\Omega}v_{\varepsilon
x}^{2}(\cdot,t)\leq\gamma^{-1}y_{\varepsilon}(t)$ for $t>0$ by (6.28). Again
restricted to times belonging to $S_{\varepsilon}$, due to (6.35) and (6.32)
this implies that

$\displaystyle-\frac{\chi_{1}}{2}\int_{\Omega}h_{1,\varepsilon}^{\prime\prime}(u_{\varepsilon})u_{\varepsilon
x}^{3}v_{\varepsilon x}$ $\displaystyle\leq$ $\displaystyle
9c_{16}\chi_{1}\varepsilon^{\theta}\int_{\Omega}u_{\varepsilon xx}^{2}$ (6.47)
$\displaystyle+\frac{1}{2}c_{6}^{4}c_{16}\chi_{1}\varepsilon^{\theta}\int_{\Omega}v_{\varepsilon
xx}^{2}+\frac{1}{2}c_{6}^{4}c_{16}c_{22}^{3}\chi_{1}\gamma^{-\frac{39}{10}}\varepsilon^{\theta}$
$\displaystyle\leq$ $\displaystyle\frac{D_{1}}{64}\int_{\Omega}u_{\varepsilon
xx}^{2}+\frac{\gamma D_{2}}{64}\int_{\Omega}v_{\varepsilon
xx}^{2}+1\qquad\mbox{for all $t\in S_{\varepsilon}$ and
}\varepsilon\in(0,\varepsilon_{\star}).$

Now by arguments quite similar to those used in (6.40)-(6.47), on splitting

$\displaystyle-\gamma\chi_{2}\int_{\Omega}\Big{(}h_{2,\varepsilon}(v_{\varepsilon})u_{\varepsilon
x}\Big{)}_{x}v_{\varepsilon xx}$ $\displaystyle=$
$\displaystyle-\gamma\chi_{2}\int_{\Omega}h_{2,\varepsilon}(v_{\varepsilon})u_{\varepsilon
xx}v_{\varepsilon
xx}+\frac{\gamma\chi_{2}}{2}\int_{\Omega}h_{2,\varepsilon}^{\prime}(v_{\varepsilon})v_{\varepsilon
x}^{2}u_{\varepsilon xx}$ (6.48)
$\displaystyle+\frac{\gamma\chi_{2}}{2}\int_{\Omega}h_{2,\varepsilon}^{\prime\prime}(v_{\varepsilon})u_{\varepsilon
x}v_{\varepsilon x}^{3},\qquad t>0,$

we may first essentially copy the reasoning from (6.41), (6.42), (6.43),
(6.44) and (6.45): Indeed, again using that $\int_{\Omega}v_{\varepsilon
x}^{2}(\cdot,t)\leq\gamma^{-1}y_{\varepsilon}(t)$ for $t>0$ we thereby obtain
that

$\displaystyle-\gamma\chi_{2}\int_{\Omega}h_{2,\varepsilon}(v_{\varepsilon})u_{\varepsilon
xx}v_{\varepsilon
xx}+\frac{\gamma\chi_{2}}{2}\int_{\Omega}h_{2,\varepsilon}^{\prime}(v_{\varepsilon})v_{\varepsilon
x}^{2}u_{\varepsilon xx}$ (6.49) $\displaystyle\leq$ $\displaystyle
c_{15}\gamma\chi_{2}\cdot\Big{\\{}\|v_{\varepsilon
x}\|_{L^{2}(\Omega)}^{\frac{2}{3}}+1\Big{\\}}\|u_{\varepsilon
xx}\|_{L^{2}(\Omega)}\Big{\\{}\|v_{\varepsilon
xx}\|_{L^{2}(\Omega)}+1\Big{\\}}$ $\displaystyle\leq$ $\displaystyle
c_{15}\gamma\chi_{2}\cdot\Big{\\{}\gamma^{-\frac{1}{3}}y_{\varepsilon}^{\frac{1}{3}}(t)+1\Big{\\}}\|u_{\varepsilon
xx}\|_{L^{2}(\Omega)}\Big{\\{}\|v_{\varepsilon
xx}\|_{L^{2}(\Omega)}+1\Big{\\}}$ $\displaystyle\leq$
$\displaystyle\frac{D_{1}}{32}\|u_{\varepsilon
xx}\|_{L^{2}(\Omega)}^{2}+\frac{8c_{15}^{2}\gamma^{2}\chi_{2}^{2}}{D_{1}}\cdot\Big{\\{}\gamma^{-\frac{1}{3}}y_{\varepsilon}^{\frac{1}{3}}(t)+1\Big{\\}}^{2}\cdot\Big{\\{}\|v_{\varepsilon
xx}\|_{L^{2}(\Omega)}+1\Big{\\}}^{2}$ $\displaystyle\leq$
$\displaystyle\frac{D_{1}}{32}\int_{\Omega}u_{\varepsilon
xx}^{2}+\frac{16c_{15}^{2}\gamma^{2}\chi_{2}^{2}}{D_{1}}\cdot\Big{\\{}\gamma^{-\frac{1}{3}}y_{\varepsilon}^{\frac{1}{3}}(t)+1\Big{\\}}^{2}\cdot\bigg{\\{}\int_{\Omega}v_{\varepsilon
xx}^{2}+1\bigg{\\}}$ $\displaystyle\leq$
$\displaystyle\frac{D_{1}}{32}\int_{\Omega}u_{\varepsilon xx}^{2}+\frac{\gamma
D_{2}}{32}\int_{\Omega}v_{\varepsilon xx}^{2}+\frac{\gamma
D_{2}}{32}\qquad\mbox{for all $t\in S_{\varepsilon}$ and
}\varepsilon\in(0,1),$

because according to (6.35) and the inequalities $\gamma\leq 1$ and
$c_{22}^{\frac{1}{3}}\gamma^{-\frac{13}{30}}\geq 1$ asserted by (6.25) and
(6.29) we have

$\displaystyle\frac{\frac{16c_{15}^{2}\gamma^{2}\chi_{2}^{2}}{D_{1}}\cdot\Big{\\{}\gamma^{-\frac{1}{3}}y_{\varepsilon}^{\frac{1}{3}}(t)+1\Big{\\}}^{2}}{\frac{\gamma
D_{2}}{32}}$ $\displaystyle=$
$\displaystyle\frac{512c_{15}^{2}\gamma\chi_{2}^{2}}{D_{1}D_{2}}\cdot\Big{\\{}\gamma^{-\frac{1}{3}}y_{\varepsilon}^{\frac{1}{3}}(t)+1\Big{\\}}^{2}$
$\displaystyle<$
$\displaystyle\frac{512c_{15}^{2}\gamma\chi_{2}^{2}}{D_{1}D_{2}}\cdot(c_{22}^{\frac{1}{3}}\gamma^{-\frac{13}{30}}+1)^{2}$
$\displaystyle\leq$
$\displaystyle\frac{2048c_{15}^{2}c_{22}^{\frac{2}{3}}\gamma^{\frac{2}{15}}\chi_{2}^{2}}{D_{1}D_{2}}$
$\displaystyle\leq$
$\displaystyle\frac{2048c_{15}^{2}c_{22}^{\frac{2}{3}}\chi_{\star}^{2}}{D_{1}D_{2}}$
$\displaystyle\leq$ $\displaystyle 1\qquad\mbox{for all $t\in S_{\varepsilon}$
and }\varepsilon\in(0,1)$

due to (6.26) and our restriction that $1\geq c_{23}^{\frac{3}{5}}$.
Likewise, proceeding as in (6.46) and (6.47) we find that since
$\int_{\Omega}u_{\varepsilon x}^{2}(\cdot,t)\leq
y_{\varepsilon}(t)<c_{22}\gamma^{-\frac{3}{10}}$ for all $t\in
S_{\varepsilon}$,

$\displaystyle\frac{\gamma\chi_{2}}{2}\int_{\Omega}h_{2,\varepsilon}^{\prime\prime}(v_{\varepsilon})u_{\varepsilon
x}v_{\varepsilon x}^{3}$ $\displaystyle\leq$ $\displaystyle
9c_{16}\gamma\chi_{2}\varepsilon^{\theta}\int_{\Omega}v_{\varepsilon
xx}^{2}+c_{6}^{4}c_{16}\gamma\chi_{2}\varepsilon^{\theta}\|u_{\varepsilon
xx}\|_{L^{2}(\Omega)}\|u_{\varepsilon x}\|_{L^{2}(\Omega)}^{3}$ (6.50)
$\displaystyle\leq$ $\displaystyle
9c_{16}\gamma\chi_{2}\varepsilon^{\theta}\int_{\Omega}v_{\varepsilon
xx}^{2}+\frac{1}{2}c_{6}^{4}c_{16}\gamma\chi_{2}\varepsilon^{\theta}\int_{\Omega}u_{\varepsilon
xx}^{2}+\frac{1}{2}c_{6}^{4}c_{16}c_{22}^{3}\gamma^{\frac{1}{10}}\chi_{2}\varepsilon^{\theta}$
$\displaystyle\leq$ $\displaystyle\frac{D_{1}}{64}\int_{\Omega}u_{\varepsilon
xx}^{2}+\frac{\gamma D_{2}}{64}\int_{\Omega}v_{\varepsilon
xx}^{2}+1\qquad\mbox{for all $t\in S_{\varepsilon}$ and
}\varepsilon\in(0,\varepsilon_{\star})$

because of (6.33).
When inserted into (6.36), in view of (6.40) and (6.48) the estimates (6.37),
(6.38), (6.39), (6.45), (6.47), (6.49) and (6.50) in summary show that as
$\gamma\leq 1$,

$\displaystyle\frac{1}{2}y_{\varepsilon}^{\prime}(t)+\frac{D_{1}}{32}\int_{\Omega}u_{\varepsilon
xx}^{2}+\frac{\gamma D_{2}}{32}\int_{\Omega}v_{\varepsilon xx}^{2}$
$\displaystyle\leq$ $\displaystyle
c_{12}\gamma^{-\frac{1}{2}}+c_{13}+c_{14}+\frac{D_{1}}{32}+1+\frac{\gamma
D_{2}}{32}+1$ (6.51) $\displaystyle\leq$ $\displaystyle
c_{17}\gamma^{-\frac{1}{2}}\qquad\mbox{for all $t\in S_{\varepsilon}$ and
}\varepsilon\in(0,\varepsilon_{\star})$

according to the definition of $c_{17}$ in (6.23). In order to turn the two
remaining integrals herein into an appropriate superlinear absorptive term, we
once more make use of (6.12), (6.30) and Young’s inequality to see that again
since $\gamma\leq 1$,

$\displaystyle y_{\varepsilon}^{\frac{5}{3}}(t)$ $\displaystyle=$
$\displaystyle\bigg{\\{}\int_{\Omega}u_{\varepsilon
x}^{2}+\gamma\int_{\Omega}v_{\varepsilon x}^{2}\bigg{\\}}^{\frac{5}{3}}$
$\displaystyle\leq$ $\displaystyle
c_{4}^{\frac{5}{3}}\cdot\bigg{\\{}\|u_{\varepsilon
xx}\|_{L^{2}(\Omega)}^{\frac{6}{5}}+1+\gamma\cdot\Big{(}\|v_{\varepsilon
xx}\|_{L^{2}(\Omega)}^{\frac{6}{5}}+1\Big{)}\bigg{\\}}^{\frac{5}{3}}$
$\displaystyle\leq$ $\displaystyle
c_{4}^{\frac{5}{3}}\cdot\Big{\\{}\|u_{\varepsilon
xx}\|_{L^{2}(\Omega)}^{\frac{6}{5}}+\gamma^{\frac{3}{5}}\|v_{\varepsilon
xx}\|_{L^{2}(\Omega)}^{\frac{6}{5}}+2\Big{\\}}^{\frac{5}{3}}$
$\displaystyle\leq$
$\displaystyle(3c_{4})^{\frac{5}{3}}\cdot\Big{\\{}\|u_{\varepsilon
xx}\|_{L^{2}(\Omega)}^{2}+\gamma\|v_{\varepsilon
xx}\|_{L^{2}(\Omega)}^{2}+2^{\frac{5}{3}}\Big{\\}}\qquad\mbox{for all
$t>T_{1}$ and all }\varepsilon\in(0,1),$

so that for all $t>T_{1}$ and any $\varepsilon\in(0,1)$,

$\displaystyle\frac{D_{1}}{32}\int_{\Omega}u_{\varepsilon xx}^{2}+\frac{\gamma
D_{2}}{32}\int_{\Omega}v_{\varepsilon xx}^{2}$ $\displaystyle\geq$
$\displaystyle\frac{\min\\{D_{1},D_{2}\\}}{32}\cdot\bigg{\\{}\int_{\Omega}u_{\varepsilon
xx}^{2}+\gamma\int_{\Omega}v_{\varepsilon xx}^{2}\bigg{\\}}$
$\displaystyle\geq$
$\displaystyle\frac{\min\\{D_{1},D_{2}\\}}{32}\cdot(3c_{4})^{-\frac{5}{3}}\cdot
y_{\varepsilon}^{\frac{5}{3}}(t)-\frac{2^{\frac{2}{3}}\cdot\min\\{D_{1},D_{2}\\}}{16}.$

In light of (6.24), (6.51) therefore entails the autonomous ODI

$\displaystyle\frac{1}{2}y_{\varepsilon}^{\prime}(t)+c_{19}y_{\varepsilon}^{\frac{5}{3}}(t)\leq
c_{20}\gamma^{-\frac{1}{2}}\qquad\mbox{for all $t\in S_{\varepsilon}$ and each
}\varepsilon\in(0,\varepsilon_{\star}).$ (6.52)

To deduce (6.34) from this, we now recall (6.31) to infer that for each
$\varepsilon\in(0,1)$ we can fix $t_{\varepsilon}\in(T_{\star}-1,T_{\star})$
such that

$\int_{\Omega}\frac{u_{\varepsilon
x}^{2}(\cdot,t_{\varepsilon})}{u_{\varepsilon}(\cdot,t_{\varepsilon})}+\rho\int_{\Omega}\frac{v_{\varepsilon
x}^{2}(\cdot,t_{\varepsilon})}{v_{\varepsilon}(\cdot,t_{\varepsilon})}\leq
2K.$ (6.53)

Through (6.15) and again (6.30), this firstly entails that

$\displaystyle\|u_{\varepsilon}(\cdot,t_{\varepsilon})\|_{L^{\infty}(\Omega)}^{2}$
$\displaystyle=$
$\displaystyle\|\sqrt{u_{\varepsilon}(\cdot,t_{\varepsilon})}\|_{L^{\infty}(\Omega)}^{4}$
$\displaystyle\leq$ $\displaystyle
c_{7}\cdot\Big{\\{}\|(\sqrt{u_{\varepsilon}(\cdot,t_{\varepsilon})})_{x}\|_{L^{2}(\Omega)}^{2}+1\Big{\\}}$
$\displaystyle=$
$\displaystyle\frac{c_{7}}{4}\int_{\Omega}\frac{u_{\varepsilon
x}^{2}(\cdot,t_{\varepsilon})}{u_{\varepsilon}(\cdot,t_{\varepsilon})}+c_{7}$
$\displaystyle\leq$ $\displaystyle\frac{c_{7}K}{2}+c_{7}\qquad\mbox{for all
}\varepsilon\in(0,1)$

and, similarly,

$\displaystyle\|v_{\varepsilon}(\cdot,t_{\varepsilon})\|_{L^{\infty}(\Omega)}^{2}\leq\frac{c_{7}}{4}\int_{\Omega}\frac{v_{\varepsilon
x}^{2}(\cdot,t_{\varepsilon})}{v_{\varepsilon}(\cdot,t_{\varepsilon})}+c_{7}\leq\frac{c_{7}K}{2\rho}+c_{7}\qquad\mbox{for
all }\varepsilon\in(0,1).$

Therefore, (6.53) secondly implies that

$\displaystyle y_{\varepsilon}(t_{\varepsilon})$ $\displaystyle\leq$
$\displaystyle\|u_{\varepsilon}(\cdot,t_{\varepsilon})\|_{L^{\infty}(\Omega)}\int_{\Omega}\frac{u_{\varepsilon
x}^{2}(\cdot,t_{\varepsilon})}{u_{\varepsilon}(\cdot,t_{\varepsilon})}+\gamma\|v_{\varepsilon}(\cdot,t_{\varepsilon})\|_{L^{\infty}(\Omega)}\int_{\Omega}\frac{v_{\varepsilon
x}^{2}(\cdot,t_{\varepsilon})}{v_{\varepsilon}(\cdot,t_{\varepsilon})}$ (6.54)
$\displaystyle\leq$ $\displaystyle\sqrt{\frac{c_{7}K}{2}+c_{7}}\cdot
2K+\gamma\cdot\sqrt{\frac{c_{7}K}{2\rho}+c_{7}}\cdot\frac{2K}{\rho}$
$\displaystyle\leq$ $\displaystyle
c_{21}+c_{21}\gamma\rho^{-\frac{3}{2}}\qquad\mbox{for all
}\varepsilon\in(0,1)$

by (6.25), for we are yet assuming that $\rho\leq 1$.
But now the precise link between $\gamma$ and $\rho$ in (6.29) ensures that

$\displaystyle\frac{c_{21}\gamma\rho^{-\frac{3}{2}}}{\frac{1}{4}c_{22}\gamma^{-\frac{3}{10}}}=\frac{4c_{21}}{c_{22}}\gamma^{\frac{13}{10}}\rho^{-\frac{3}{2}}=\frac{4c_{21}c_{23}^{\frac{13}{10}}}{c_{22}}\leq
1$

due to (6.25), so that since (6.25) moreover entails that

$\displaystyle\frac{c_{21}}{\frac{1}{4}c_{22}\gamma^{-\frac{3}{10}}}\leq\frac{4c_{21}}{c_{22}}\leq
1,$

from (6.54) we infer that

$y_{\varepsilon}(t_{\varepsilon})\leq\frac{1}{2}c_{22}\gamma^{-\frac{3}{10}}\qquad\mbox{for
all }\varepsilon\in(0,1).$ (6.55)

This especially guarantees that $t_{\varepsilon}$ belongs to $S_{\varepsilon}$
and that hence

$\displaystyle T_{\varepsilon}:=\sup\Big{\\{}\widehat{T}>t_{\varepsilon}\
\Big{|}\ [t_{\varepsilon},\widehat{T}]\subset S_{\varepsilon}\Big{\\}}$

is a well-defined element of $(t_{\varepsilon},\infty]$ for all
$\varepsilon\in(0,1)$. According to (6.52), however, for each
$\varepsilon\in(0,\varepsilon_{\star})$ we actually must have
$T_{\varepsilon}=\infty$: Otherwise, namely, the definition of
$S_{\varepsilon}$ would entail that $(t_{\varepsilon},T_{\varepsilon})\subset
S_{\varepsilon}$ but

$y_{\varepsilon}(T_{\varepsilon})=c_{22}\gamma^{-\frac{3}{10}},$ (6.56)

while since $\overline{y}(t):=\frac{1}{2}c_{22}\gamma^{-\frac{3}{10}}$, $t\geq
t_{\varepsilon}$, satisfies

$\displaystyle\frac{1}{2}\overline{y}^{\prime}(t)+c_{19}\overline{y}^{\frac{5}{3}}(t)-c_{20}\gamma^{-\frac{1}{2}}=(2^{-\frac{5}{3}}c_{19}c_{22}^{\frac{5}{3}}-c_{20})\cdot\gamma^{-\frac{1}{2}}\geq
0\qquad\mbox{for all }t>t_{\varepsilon}$

as a further consequence of (6.25), a comparison argument based on (6.55)
would show that $y_{\varepsilon}(t)\leq\overline{y}(t)$ for all
$t\in[t_{\varepsilon},T_{\varepsilon}]$ and thereby contradict (6.56). This
confirms that indeed $[t_{\varepsilon},\infty)\subset S_{\varepsilon}$ for all
$\varepsilon\in(0,\varepsilon_{\star})$ and that thus (6.34) results from the
fact that $t_{\varepsilon}<T_{\star}$ for any such $\varepsilon$. In view of
(6.30) and e.g. (6.11), we therefore conclude that both (6.6) and (6.7) are
valid whenever $0<\chi_{1}\leq\chi_{2}<\chi_{\star}$, whereupon going back to
(6.51) we infer by integration that also (6.8) holds for all such $\chi_{1}$
and $\chi_{2}$.
According to an evident symmetry property of the presented reasoning, however,
in the case when conversely $\chi_{2}\leq\chi_{1}$ the above result can be
derived in much the same manner, essentially by exchanging the roles of
$u_{\varepsilon}$ and $v_{\varepsilon}$. $\Box$ By once more recalling the
convergence properties obtained in Section 4, from Lemma 6.2 we readily derive
the following conclusion, inter alia asserting eventual continuity and
$H^{1}$-boundedness of the limit couple from Lemma 4.5.

###### Corollary 6.3

Let $D_{i}>0,a_{i}>0,\lambda_{i}>0$ and $n_{i}\in[1,2]$ for $i\in\\{1,2\\}$,
and let $\chi_{1}\in(0,\chi_{\star})$ and $\chi_{2}\in(0,\chi_{\star})$ with
$\chi_{\star}>0$ as given by Lemma 6.2. Then whenever (1.5) holds, there
exists $T_{\star}=T_{\star}(\chi_{1},\chi_{2},u_{0},v_{0})>0$ such that with
$(\varepsilon_{j})_{j\in\mathbb{N}}\subset(0,1)$ and $N\subset(0,\infty)$
taken from Lemma 4.5 we have

$u_{\varepsilon}(\cdot,t)\to u(\cdot,t)\quad\mbox{and}\quad
v_{\varepsilon}(\cdot,t)\to v(\cdot,t)\quad\mbox{in
}L^{\infty}(\Omega)\qquad\mbox{for all }t\in(0,\infty)\setminus N$ (6.57)

as well as

$\int_{t_{1}}^{t_{2}}\int_{\Omega}u_{\varepsilon
x}^{2}\to\int_{t_{1}}^{t_{2}}\int_{\Omega}u_{x}^{2}\quad\mbox{and}\quad\int_{t_{1}}^{t_{2}}\int_{\Omega}v_{\varepsilon
x}^{2}\to\int_{t_{1}}^{t_{2}}\int_{\Omega}v_{x}^{2}\qquad\mbox{for all
$t_{1}>T_{\star}$ and any }t_{2}>t_{1}$ (6.58)

as $\varepsilon=\varepsilon_{j}\searrow 0$. Moreover, both $u$ and $v$ belong
to $C^{0}(\overline{\Omega}\times(T_{\star},\infty))$ and satisfy
$u(\cdot,t)\in W^{1,2}(\Omega)$ and $v(\cdot,t)\in W^{1,2}(\Omega)$ for all
$t>T_{\star}$, and one can find $C>0$ such that

$\|u(\cdot,t)\|_{W^{1,2}(\Omega)}+\|v(\cdot,t)\|_{W^{1,2}(\Omega)}\leq
C\qquad\mbox{for all }t>T_{\star}.$ (6.59)

Proof. According to Lemma 6.2, for fixed $u_{0}$ and $v_{0}$ we can find
$T_{\star}>0$, $c_{1}>0$, $c_{2}>0$ and $\varepsilon_{\star}\in(0,1)$ such
that for any $\varepsilon\in(0,\varepsilon_{\star})$,

$\|u_{\varepsilon}(\cdot,t)\|_{W^{1,2}(\Omega)}+\|v_{\varepsilon}(\cdot,t)\|_{W^{1,2}(\Omega)}\leq
c_{1}\qquad\mbox{for all }t>T_{\star}$ (6.60)

and

$\int_{T_{\star}}^{T}\|u_{\varepsilon}(\cdot,t)\|_{W^{2,2}(\Omega)}^{2}dt+\int_{T_{\star}}^{T}\|v_{\varepsilon}(\cdot,t)\|_{W^{2,2}(\Omega)}^{2}dt\leq
c_{2}\cdot(T+1)\qquad\mbox{for all }T>T_{\star}.$ (6.61)

As (6.60) warrants relative compactness of
$(u_{\varepsilon}(\cdot,t))_{\varepsilon\in(0,\varepsilon_{\star})}$ and of
$(v_{\varepsilon}(\cdot,t))_{\varepsilon\in(0,\varepsilon_{\star})}$ with
respect to the weak topology in $W^{1,2}(\Omega)$ for all $t>T_{\star}$, it is
thus clear from Lemma 4.5 that as $\varepsilon=\varepsilon_{j}\searrow 0$,

$u_{\varepsilon}(\cdot,t)\rightharpoonup u(\cdot,t)\quad\mbox{and}\quad
v_{\varepsilon}(\cdot,t)\rightharpoonup v(\cdot,t)\quad\mbox{in
}W^{1,2}(\Omega)\qquad\mbox{for all }t\in(T_{\star},\infty)\setminus N.$
(6.62)

and thereby especially entails (6.57) by compactness of the embedding
$W^{1,2}(\Omega)\hookrightarrow L^{\infty}(\Omega)$. Since thus
$(u(\cdot,t))_{t\in(T_{\star},\infty)\setminus N}$ and
$(v(\cdot,t))_{t\in(T_{\star},\infty)\setminus N}$ are bounded in
$W^{1,2}(\Omega)$ due to (6.62) and (6.60), and since from Lemma 4.5 we
already know that both $u$ and $v$ belong to
$C^{0}_{w}([0,\infty);L^{1}(\Omega))$, it can therefore easily be verified
that actually $u$ and $v$ are continuous throughout $(T_{\star},\infty)$ as
$W^{1,2}(\Omega)$-valued functions with respect to the weak topology therein.
Again through compactness of the embedding $W^{1,2}(\Omega)\hookrightarrow
L^{\infty}(\Omega)$, this entails that indeed $u$ and $v$ are contained in
$C^{0}((T_{\star},\infty);C^{0}(\overline{\Omega}))$ and satisfy (6.59).
Apart from that, for all $T>T_{\star}$ (6.61) ensures boundedness of the
families $(u_{\varepsilon})_{\varepsilon\in(0,\varepsilon_{\star})}$ and
$(v_{\varepsilon})_{\varepsilon\in(0,\varepsilon_{\star})}$ in
$L^{2}((T_{\star},T);W^{2,2}(\Omega))$, so that recalling the time regularity
properties from Lemma 4.4 we may once more invoke the Aubin-Lions lemma to
conclude that $u_{\varepsilon}\to u$ and $v_{\varepsilon}\to v$ in
$L^{2}((T_{\star},t);W^{1,2}(\Omega))$, and that thus in particular (6.58)
holds. $\Box$

##  7 Toward verifying consistency with a genuine entropy structure

Now having at hand the eventual regularity properties gained above, for
establishing our main results on stabilization in the flavor of Theorem 1.2
and Theorem 1.3 it will be sufficient to make sure that the genuine gradient-
like structures related to the functionals in (1.12) and (1.14) find some
appropriate rigorous counterpart in the approximate problems (1.6)-(1.7). This
short section provides the fundament therefor by recording results of
associated testing procedures, which will in particular foreshadow our final
selections of the key parameters $n_{1}$ and $n_{2}$.

###### Lemma 7.1

Let $D_{i}>0,a_{i}>0,\lambda_{i}>0$ and $n_{i}\in[1,2]$ for $i\in\\{1,2\\}$,
and assume (1.5).
i) If $n_{1}=2$, then for all $\varepsilon\in(0,1)$,

$\displaystyle\frac{d}{dt}\bigg{\\{}-\int_{\Omega}\ln
u_{\varepsilon}+\frac{\varepsilon}{6}\int_{\Omega}\frac{1}{u_{\varepsilon}^{2}}\bigg{\\}}$
$\displaystyle+$ $\displaystyle\varepsilon\int_{\Omega}u_{\varepsilon
xx}^{2}+D_{1}\int_{\Omega}\frac{u_{\varepsilon
x}^{2}}{u_{\varepsilon}^{2}}+D_{1}\varepsilon\int_{\Omega}\frac{u_{\varepsilon
x}^{2}}{u_{\varepsilon}^{4}}$ (7.1) $\displaystyle+$
$\displaystyle\varepsilon^{\frac{\alpha}{2}}\int_{\Omega}u_{\varepsilon}^{-\alpha-2}u_{\varepsilon
x}^{2}+\varepsilon^{\frac{\alpha+2}{2}}\int_{\Omega}u_{\varepsilon}^{-\alpha-4}u_{\varepsilon
x}^{2}$ $\displaystyle=$
$\displaystyle\chi_{1}\int_{\Omega}\frac{1}{u_{\varepsilon}}u_{\varepsilon
x}v_{\varepsilon
x}-\lambda_{1}|\Omega|+\int_{\Omega}u_{\varepsilon}-a_{1}\int_{\Omega}v_{\varepsilon}\qquad\mbox{for
all }t>0.$

ii) If $n_{2}=2$, then for all $\varepsilon\in(0,1)$,

$\displaystyle\frac{d}{dt}\bigg{\\{}-\int_{\Omega}\ln
v_{\varepsilon}+\frac{\varepsilon}{6}\int_{\Omega}\frac{1}{v_{\varepsilon}^{2}}\bigg{\\}}$
$\displaystyle+$ $\displaystyle\varepsilon\int_{\Omega}v_{\varepsilon
xx}^{2}+D_{2}\int_{\Omega}\frac{v_{\varepsilon
x}^{2}}{v_{\varepsilon}^{2}}+D_{2}\varepsilon\int_{\Omega}\frac{v_{\varepsilon
x}^{2}}{v_{\varepsilon}^{4}}$ (7.2) $\displaystyle+$
$\displaystyle\varepsilon^{\frac{\alpha}{2}}\int_{\Omega}v_{\varepsilon}^{-\alpha-2}v_{\varepsilon
x}^{2}+\varepsilon^{\frac{\alpha+2}{2}}\int_{\Omega}v_{\varepsilon}^{-\alpha-4}v_{\varepsilon
x}^{2}$ $\displaystyle=$
$\displaystyle-\chi_{2}\int_{\Omega}\frac{1}{v_{\varepsilon}}u_{\varepsilon
x}v_{\varepsilon
x}-\lambda_{2}|\Omega|+\int_{\Omega}v_{\varepsilon}+a_{2}\int_{\Omega}u_{\varepsilon}\qquad\mbox{for
all }t>0.$

iii) In the case $n_{2}=1$, for all $\varepsilon\in(0,1)$ we have

$\displaystyle\frac{d}{dt}\bigg{\\{}\frac{1}{2}\int_{\Omega}v_{\varepsilon}^{2}+\frac{\varepsilon}{2}\int_{\Omega}\frac{1}{v_{\varepsilon}}\bigg{\\}}$
$\displaystyle+$
$\displaystyle\varepsilon\int_{\Omega}v_{\varepsilon}v_{\varepsilon
xx}^{2}+D_{2}\int_{\Omega}v_{\varepsilon
x}^{2}+D_{2}\varepsilon\int_{\Omega}\frac{v_{\varepsilon
x}^{2}}{v_{\varepsilon}^{3}}$ (7.3) $\displaystyle+$
$\displaystyle\varepsilon^{\frac{\alpha}{2}}\int_{\Omega}v_{\varepsilon}^{-\alpha}v_{\varepsilon
x}^{2}+\varepsilon^{\frac{\alpha+2}{2}}\int_{\Omega}v_{\varepsilon}^{-\alpha-3}v_{\varepsilon
x}^{2}$ $\displaystyle=$
$\displaystyle-\chi_{2}\int_{\Omega}v_{\varepsilon}u_{\varepsilon
x}v_{\varepsilon
x}+\lambda_{2}\int_{\Omega}v_{\varepsilon}^{2}-\int_{\Omega}v_{\varepsilon}^{3}-a_{2}\int_{\Omega}u_{\varepsilon}v_{\varepsilon}^{2}$
$\displaystyle-\varepsilon\int_{\Omega}\frac{2v_{\varepsilon}^{2}+3v_{\varepsilon}}{6v_{\varepsilon}^{2}+2\varepsilon}(\lambda_{2}-v_{\varepsilon}-a_{2}u_{\varepsilon})\qquad\mbox{for
all }t>0.$

Proof. i) Writing $\ell(s):=-\ln s+\frac{\varepsilon}{6}\cdot\frac{1}{s^{2}}$
for $s>0$ and fixed $\varepsilon\in(0,1)$, we have
$\ell^{\prime}(s)=-\frac{1}{s}-\frac{\varepsilon}{3s^{3}}=-\frac{3s^{2}+\varepsilon}{3s^{3}}$
and
$\ell^{\prime\prime}(s)=\frac{1}{s^{2}}+\frac{\varepsilon}{s^{4}}=\frac{s^{2}+\varepsilon}{s^{4}}$
for all $s>0$, so that using the first equation in (1.6) we obtain

$\displaystyle\frac{d}{dt}\int_{\Omega}\ell(u_{\varepsilon})$ $\displaystyle=$
$\displaystyle-\varepsilon\int_{\Omega}\ell^{\prime}(u_{\varepsilon})\cdot\Big{(}\frac{u_{\varepsilon}^{4}}{u_{\varepsilon}^{2}+\varepsilon}u_{\varepsilon
xxx}\Big{)}_{x}+\varepsilon^{\frac{\alpha}{2}}\int_{\Omega}\ell^{\prime}(u_{\varepsilon})\cdot(u_{\varepsilon}^{-\alpha}u_{\varepsilon
x})_{x}$
$\displaystyle+D_{1}\int_{\Omega}\ell^{\prime}(u_{\varepsilon})u_{\varepsilon
xx}-\chi_{1}\int_{\Omega}\ell^{\prime}(u_{\varepsilon})\cdot\Big{(}\frac{u_{\varepsilon}^{3}}{u_{\varepsilon}^{2}+\varepsilon}v_{\varepsilon
x}\Big{)}_{x}$
$\displaystyle+\int_{\Omega}\ell^{\prime}(u_{\varepsilon})\cdot\frac{3u_{\varepsilon}^{3}}{3u_{\varepsilon}^{2}+\varepsilon}\cdot(\lambda_{1}-u_{\varepsilon}+a_{1}v_{\varepsilon})$
$\displaystyle=$
$\displaystyle\varepsilon\int_{\Omega}\ell^{\prime\prime}(u_{\varepsilon})\cdot\frac{u_{\varepsilon}^{4}}{u_{\varepsilon}^{2}+\varepsilon}u_{\varepsilon
x}u_{\varepsilon
xxx}-\varepsilon^{\frac{\alpha}{2}}\int_{\Omega}\ell^{\prime\prime}(u_{\varepsilon})u_{\varepsilon}^{-\alpha}u_{\varepsilon
x}^{2}$ $\displaystyle-
D_{1}\int_{\Omega}\ell^{\prime\prime}(u_{\varepsilon})u_{\varepsilon
x}^{2}+\chi_{1}\int_{\Omega}\ell^{\prime\prime}(u_{\varepsilon})\cdot\frac{u_{\varepsilon}^{3}}{u_{\varepsilon}^{2}+\varepsilon}u_{\varepsilon
x}v_{\varepsilon x}$
$\displaystyle+\int_{\Omega}\ell^{\prime}(u_{\varepsilon})\cdot\frac{3u_{\varepsilon}^{3}}{3u_{\varepsilon}^{2}+\varepsilon}\cdot(\lambda_{1}-u_{\varepsilon}+a_{1}v_{\varepsilon})$
$\displaystyle=$ $\displaystyle\varepsilon\int_{\Omega}u_{\varepsilon
x}u_{\varepsilon
xxx}-\varepsilon^{\frac{\alpha}{2}}\int_{\Omega}(u_{\varepsilon}^{-\alpha-2}+\varepsilon
u_{\varepsilon}^{-\alpha-4})u_{\varepsilon x}^{2}$ $\displaystyle-
D_{1}\int_{\Omega}\Big{(}\frac{1}{u_{\varepsilon}^{2}}+\frac{\varepsilon}{u_{\varepsilon}^{4}}\Big{)}u_{\varepsilon
x}^{2}+\chi_{1}\int_{\Omega}\frac{1}{u_{\varepsilon}}u_{\varepsilon
x}v_{\varepsilon x}$
$\displaystyle-\int_{\Omega}(\lambda_{1}-u_{\varepsilon}+a_{1}v_{\varepsilon})\qquad\mbox{for
all }t>0.$

As clearly $\varepsilon\int_{\Omega}u_{\varepsilon x}u_{\varepsilon
xxx}=-\varepsilon\int_{\Omega}u_{\varepsilon xx}^{2}$ for all $t>0$, this is
equivalent to (7.1).
ii) This part analogously follows from the second equation in (1.6).
iii) We now rather let $\ell(s):=\frac{1}{2}s^{2}+\frac{\varepsilon}{2s}$ for
$s>0$ and $\varepsilon\in(0,1)$, and observing that then
$\ell^{\prime}(s)=s-\frac{\varepsilon}{2s^{2}}=\frac{2s^{3}-\varepsilon}{2s^{2}}$
and
$\ell^{\prime\prime}(s)=1+\frac{\varepsilon}{s^{3}}=\frac{s^{3}+\varepsilon}{s^{3}}$
for all $s>0$, we use (1.6)-(1.7) to compute

$\displaystyle\frac{d}{dt}\int_{\Omega}\ell(v_{\varepsilon})$ $\displaystyle=$
$\displaystyle\varepsilon\int_{\Omega}\ell^{\prime\prime}(v_{\varepsilon})\cdot\frac{v_{\varepsilon}^{4}}{v_{\varepsilon}^{3}+\varepsilon}v_{\varepsilon
x}v_{\varepsilon
xxx}-\varepsilon^{\frac{\alpha}{2}}\int_{\Omega}\ell^{\prime\prime}(v_{\varepsilon})v_{\varepsilon}^{-\alpha}v_{\varepsilon
x}^{2}$ $\displaystyle-
D_{2}\int_{\Omega}\ell^{\prime\prime}(v_{\varepsilon})v_{\varepsilon
x}^{2}-\chi_{2}\int_{\Omega}\ell^{\prime\prime}(v_{\varepsilon})\cdot\frac{v_{\varepsilon}^{4}}{v_{\varepsilon}^{3}+\varepsilon}u_{\varepsilon
x}v_{\varepsilon x}$
$\displaystyle+\int_{\Omega}\ell^{\prime}(v_{\varepsilon})\cdot\frac{3v_{\varepsilon}^{3}}{3v_{\varepsilon}^{2}+\varepsilon}\cdot(\lambda_{2}-v_{\varepsilon}-a_{2}u_{\varepsilon})$
$\displaystyle=$
$\displaystyle\varepsilon\int_{\Omega}v_{\varepsilon}v_{\varepsilon
x}v_{\varepsilon
xxx}-\varepsilon^{\frac{\alpha}{2}}\int_{\Omega}(v_{\varepsilon}^{-\alpha}+\varepsilon
v_{\varepsilon}^{-\alpha-3})v_{\varepsilon x}^{2}$ $\displaystyle-
D_{2}\int_{\Omega}\Big{(}1+\frac{\varepsilon}{v_{\varepsilon}^{3}}\Big{)}v_{\varepsilon
x}^{2}-\chi_{2}\int_{\Omega}v_{\varepsilon}u_{\varepsilon x}v_{\varepsilon x}$
$\displaystyle+\int_{\Omega}\frac{3v_{\varepsilon}(2v_{\varepsilon}^{3}-\varepsilon)}{2\cdot(3v_{\varepsilon}^{2}+\varepsilon)}\cdot(\lambda_{2}-v_{\varepsilon}-a_{1}u_{\varepsilon})\qquad\mbox{for
all }t>0.$

Since

$\displaystyle\varepsilon\int_{\Omega}v_{\varepsilon}v_{\varepsilon
x}v_{\varepsilon xxx}=-\varepsilon\int_{\Omega}v_{\varepsilon}v_{\varepsilon
xx}^{2}-\varepsilon\int_{\Omega}v_{\varepsilon x}^{2}v_{\varepsilon
xx}=-\varepsilon\int_{\Omega}v_{\varepsilon}v_{\varepsilon
xx}^{2}\qquad\mbox{for all }t>0,$

and since

$\displaystyle\frac{3s(2s^{3}-\varepsilon)}{2\cdot(3s^{2}+\varepsilon)}=s^{2}-\varepsilon\cdot\frac{2s^{2}+3s}{6s^{2}+2\varepsilon}\qquad\mbox{for
all }s>0,$

this yields (7.3). $\Box$

##  8 The case $\lambda_{2}>a_{2}\lambda_{1}$. Proof of Theorem 1.2

In our convergence argument concentrating on the situation of Theorem 1.2
first, in view of (1.12) our design of an approximate variant of the
functional therein will be based on parts i) and ii) of Lemma 7.1, thus
suggesting to fix $n_{1}=n_{2}=2$ henceforth in this case. For our
construction here and also for later reference, let us separately state some
elementary properties of the nonlinear map arising in both integrals in
(1.12).

###### Lemma 8.1

Let $\xi_{\star}>0$ and

$\phi_{\xi_{\star}}(\xi):=\xi-\xi_{\star}-\xi_{\star}\ln\frac{\xi}{\xi_{\star}},\qquad\xi>0.$
(8.1)

Then $\phi_{\xi_{\star}}$ is positive on
$(0,\infty)\setminus\\{\xi_{\star}\\}$ with
$\phi_{\xi_{\star}}(\xi_{\star})=0$, and furthermore

$\phi_{\xi_{\star}}(\xi)\leq\frac{2}{\xi_{\star}}\cdot(\xi-\xi_{\star})^{2}\qquad\mbox{for
all }\xi\geq\frac{\xi_{\star}}{2}.$ (8.2)

Proof. Since $\phi_{\xi_{\star}}^{\prime}(\xi)=1-\frac{\xi_{\star}}{\xi}$ and
$\phi_{\xi_{\star}}^{\prime\prime}(\xi)=\frac{\xi_{\star}}{\xi^{2}}$ for all
$\xi>0$, it is clear that the zero $\xi_{\star}$ of $\phi_{\xi_{\star}}$ is a
strict and global minimum point of $\phi_{\xi_{\star}}$, and that moreover

$\displaystyle\phi_{\xi_{\star}}(\xi)\leq\sup_{\eta\geq\frac{\xi_{\star}}{2}}|\phi_{\xi_{\star}}^{\prime\prime}(\eta)|\cdot\frac{(\xi-\xi_{\star})^{2}}{2}=\frac{2}{\xi_{\star}}\cdot(\xi-\xi_{\star})^{2}$

for all $\xi\geq\frac{\xi_{\star}}{2}$. $\Box$ Then combining lemma 7.1 with
(1.12) suggests to generalize the entropy structure in question as follows.

###### Lemma 8.2

Let $n_{1}=n_{2}=2$, and suppose that $D_{i}>0,a_{i}>0,\lambda_{i}>0$ and
$\chi_{i}>0$ for $i\in\\{1,2\\}$, with

$\lambda_{2}>a_{2}\lambda_{1}.$ (8.3)

Then writing

$A:=\frac{a_{1}}{a_{2}}$ (8.4)

and taking $u_{\star}>0$ and $v_{\star}>0$ from (1.11), for

${\mathcal{E}}_{1,\varepsilon}(t):=\int_{\Omega}\phi_{u_{\star}}(u_{\varepsilon}(\cdot,t))+\frac{u_{\star}\varepsilon}{6}\int_{\Omega}\frac{1}{u_{\varepsilon}^{2}(\cdot,t)}+A\int_{\Omega}\phi_{v_{\star}}(v_{\varepsilon}(\cdot,t))+\frac{Av_{\star}\varepsilon}{6}\int_{\Omega}\frac{1}{v_{\varepsilon}^{2}(\cdot,t)},\qquad
t\geq 0,\ \varepsilon\in(0,1),$ (8.5)

with $\phi_{\xi_{\star}}$ taken from (8.1) for $\xi>0$, we have

$\displaystyle\frac{d}{dt}{\mathcal{E}}_{1,\varepsilon}(t)$ $\displaystyle+$
$\displaystyle\bigg{\\{}\frac{D_{1}u_{\star}}{2}-\frac{A\chi_{2}^{2}v_{\star}}{2D_{2}}\|u_{\varepsilon}(\cdot,t)\|_{L^{\infty}(\Omega)}^{2}\bigg{\\}}\cdot\int_{\Omega}\frac{u_{\varepsilon
x}^{2}}{u_{\varepsilon}^{2}}$ (8.6) $\displaystyle+$
$\displaystyle\bigg{\\{}\frac{AD_{2}v_{\star}}{2}-\frac{\chi_{1}^{2}u_{\star}}{2D_{1}}\|v_{\varepsilon}(\cdot,t)\|_{L^{\infty}(\Omega)}^{2}\bigg{\\}}\cdot\int_{\Omega}\frac{v_{\varepsilon
x}^{2}}{v_{\varepsilon}^{2}}$ $\displaystyle+$
$\displaystyle\int_{\Omega}(u_{\varepsilon}(\cdot,t)-u_{\star})^{2}+A\int_{\Omega}(v_{\varepsilon}(\cdot,t)-v_{\star})^{2}$
$\displaystyle+$ $\displaystyle
u_{\star}\varepsilon^{\frac{\alpha+2}{2}}\int_{\Omega}u_{\varepsilon}^{-\alpha-4}u_{\varepsilon
x}^{2}+Av_{\star}\varepsilon^{\frac{\alpha+2}{2}}\int_{\Omega}v_{\varepsilon}^{-\alpha-4}v_{\varepsilon
x}^{2}$ $\displaystyle\leq$
$\displaystyle\frac{1+a_{1}}{2\sqrt{3}}\cdot\sqrt{\varepsilon}\int_{\Omega}u_{\varepsilon}+\frac{A}{2\sqrt{3}}\cdot\sqrt{\varepsilon}\int_{\Omega}v_{\varepsilon}\qquad\mbox{for
all $t>0$ and }\varepsilon\in(0,1).$

Proof. We take an appropriate linear combination of the inequalities and
identities provided by Lemma 2.2 and Lemma 7.1 i) and ii) to see that for all
$\varepsilon\in(0,1)$,

$\displaystyle\frac{d}{dt}{\mathcal{E}}_{1,\varepsilon}(t)$ $\displaystyle+$
$\displaystyle u_{\star}\varepsilon\int_{\Omega}u_{\varepsilon
xx}^{2}+u_{\star}\varepsilon^{\frac{\alpha}{2}}\int_{\Omega}u_{\varepsilon}^{-\alpha-2}u_{\varepsilon
x}^{2}+u_{\star}\varepsilon^{\frac{\alpha+2}{2}}\int_{\Omega}u_{\varepsilon}^{-\alpha-4}u_{\varepsilon
x}^{2}$ (8.7) $\displaystyle+$ $\displaystyle
D_{1}u_{\star}\int_{\Omega}\frac{u_{\varepsilon
x}^{2}}{u_{\varepsilon}^{2}}+D_{1}u_{\star}\varepsilon\int_{\Omega}\frac{u_{\varepsilon
x}^{2}}{u_{\varepsilon}^{4}}$ $\displaystyle+$ $\displaystyle
Av_{\star}\varepsilon\int_{\Omega}v_{\varepsilon
xx}^{2}+Av_{\star}\varepsilon^{\frac{\alpha}{2}}\int_{\Omega}v_{\varepsilon}^{-\alpha-2}v_{\varepsilon
x}^{2}+Av_{\star}\varepsilon^{\frac{\alpha+2}{2}}\int_{\Omega}v_{\varepsilon}^{-\alpha-4}v_{\varepsilon
x}^{2}$ $\displaystyle+$ $\displaystyle
AD_{2}v_{\star}\int_{\Omega}\frac{v_{\varepsilon
x}^{2}}{v_{\varepsilon}^{2}}+AD_{2}v_{\star}\varepsilon\int_{\Omega}\frac{v_{\varepsilon
x}^{2}}{v_{\varepsilon}^{4}}$ $\displaystyle\leq$
$\displaystyle\Big{(}\lambda_{1}+\frac{\sqrt{\varepsilon}}{2\sqrt{3}}\Big{)}\int_{\Omega}u_{\varepsilon}-\int_{\Omega}u_{\varepsilon}^{2}+a_{1}\int_{\Omega}u_{\varepsilon}v_{\varepsilon}$
$\displaystyle+A\cdot\Big{(}\lambda_{2}+\frac{\sqrt{\varepsilon}}{2\sqrt{3}}\Big{)}\int_{\Omega}v_{\varepsilon}-A\int_{\Omega}v_{\varepsilon}^{2}-Aa_{2}\int_{\Omega}u_{\varepsilon}v_{\varepsilon}+\frac{Aa_{2}\sqrt{\varepsilon}}{2\sqrt{3}}\int_{\Omega}u_{\varepsilon}$
$\displaystyle+\chi_{1}u_{\star}\int_{\Omega}\frac{1}{u_{\varepsilon}}u_{\varepsilon
x}v_{\varepsilon
x}-\lambda_{1}u_{\star}|\Omega|+u_{\star}\int_{\Omega}u_{\varepsilon}-a_{1}u_{\star}\int_{\Omega}v_{\varepsilon}$
$\displaystyle-A\chi_{2}v_{\star}\int_{\Omega}\frac{1}{v_{\varepsilon}}u_{\varepsilon
x}v_{\varepsilon
x}-A\lambda_{2}v_{\star}|\Omega|+Av_{\star}\int_{\Omega}v_{\varepsilon}+Aa_{2}v_{\star}\int_{\Omega}u_{\varepsilon}\qquad\mbox{for
all }t>0.$

Here using Young’s inequality, we obtain that

$\displaystyle\chi_{1}u_{\star}\int_{\Omega}\frac{1}{u_{\varepsilon}}u_{\varepsilon
x}v_{\varepsilon x}$ $\displaystyle\leq$
$\displaystyle\frac{D_{1}u_{\star}}{2}\int_{\Omega}\frac{u_{\varepsilon
x}^{2}}{u_{\varepsilon}^{2}}+\frac{\chi_{1}^{2}u_{\star}}{2D_{1}}\int_{\Omega}v_{\varepsilon
x}^{2}$ $\displaystyle\leq$
$\displaystyle\frac{D_{1}u_{\star}}{2}\int_{\Omega}\frac{u_{\varepsilon
x}^{2}}{u_{\varepsilon}^{2}}+\frac{\chi_{1}^{2}u_{\star}}{2D_{1}}\|v_{\varepsilon}\|_{L^{\infty}(\Omega)}^{2}\int_{\Omega}\frac{v_{\varepsilon
x}^{2}}{v_{\varepsilon}^{2}}$

and, similarly,

$\displaystyle-A\chi_{2}v_{\star}\int_{\Omega}\frac{1}{v_{\varepsilon}}u_{\varepsilon
x}v_{\varepsilon
x}\leq\frac{AD_{2}v_{\star}}{2}\int_{\Omega}\frac{v_{\varepsilon
x}^{2}}{v_{\varepsilon}^{2}}+\frac{A\chi_{2}^{2}v_{\varepsilon}}{2D_{2}}\|u_{\varepsilon}\|_{L^{\infty}(\Omega)}^{2}\int_{\Omega}\frac{u_{\varepsilon
x}^{2}}{u_{\varepsilon}^{2}}$

for all $t>0$. Therefore, after neglecting some signed summands and
rearranging we infer from (8.7) that

$\displaystyle\frac{d}{dt}{\mathcal{E}}_{1,\varepsilon}(t)$ $\displaystyle+$
$\displaystyle\bigg{\\{}\frac{D_{1}u_{\star}}{2}-\frac{A\chi_{2}^{2}v_{\star}}{2D_{2}}\|u_{\varepsilon}\|_{L^{\infty}(\Omega)}^{2}\bigg{\\}}\cdot\int_{\Omega}\frac{u_{\varepsilon
x}^{2}}{u_{\varepsilon}^{2}}$ (8.8) $\displaystyle+$
$\displaystyle\bigg{\\{}\frac{AD_{2}v_{\star}}{2}-\frac{\chi_{1}^{2}u_{\star}}{2D_{1}}\|v_{\varepsilon}\|_{L^{\infty}(\Omega)}^{2}\bigg{\\}}\cdot\int_{\Omega}\frac{v_{\varepsilon
x}^{2}}{v_{\varepsilon}^{2}}$ $\displaystyle+$ $\displaystyle
u_{\star}\varepsilon^{\frac{\alpha+2}{2}}\int_{\Omega}u_{\varepsilon}^{-\alpha-4}u_{\varepsilon
x}^{2}+Av_{\star}\varepsilon^{\frac{\alpha+2}{2}}\int_{\Omega}v_{\varepsilon}^{-\alpha-4}v_{\varepsilon
x}^{2}$ $\displaystyle\leq$ $\displaystyle
I_{\varepsilon}(t):=-\int_{\Omega}u_{\varepsilon}^{2}+(\lambda_{1}+u_{\star}+Aa_{2}v_{\star})\int_{\Omega}u_{\varepsilon}-\lambda_{1}u_{\star}|\Omega|$
$\displaystyle-A\int_{\Omega}v_{\varepsilon}^{2}+(A\lambda_{2}-a_{1}u_{\star}+Av_{\star})\int_{\Omega}v_{\varepsilon}-A\lambda_{2}v_{\star}|\Omega|$
$\displaystyle+(a_{1}-Aa_{2})\int_{\Omega}u_{\varepsilon}v_{\varepsilon}$
$\displaystyle+\frac{1+Aa_{2}}{2\sqrt{3}}\cdot\sqrt{\varepsilon}\int_{\Omega}u_{\varepsilon}+\frac{A}{2\sqrt{3}}\cdot\sqrt{\varepsilon}\int_{\Omega}v_{\varepsilon}\qquad\mbox{for
all }t>0.$

We now make use of our definition (8.4) of $A$, which along with (1.11) namely
ensures the precise identities

$\displaystyle\lambda_{1}+u_{\star}+Aa_{2}v_{\star}=\lambda_{1}+u_{\star}+a_{1}v_{\star}=2u_{\star}\quad\mbox{and}\quad
A\lambda_{2}-a_{1}u_{\star}+Av_{\star}=2Av_{\star}$

as well as

$\displaystyle
u_{\star}^{2}+Av_{\star}^{2}-\lambda_{1}u_{\star}-A\lambda_{2}v_{\star}=\frac{a_{2}u_{\star}(u_{\star}-\lambda_{1})+a_{1}v_{\star}(v_{\star}-\lambda_{2})}{a_{2}}=\frac{a_{2}u_{\star}\cdot
a_{1}v_{\star}+a_{1}v_{\star}\cdot(-a_{2}u_{\star})}{a_{2}}=0$

and clearly also $a_{1}-Aa_{2}=0$. Accordingly, in (8.8) we may simplify

$\displaystyle
I_{\varepsilon}(t)=-\int_{\Omega}(u_{\varepsilon}-u_{\star})^{2}-A\int_{\Omega}(v_{\varepsilon}-v_{\star})^{2}+\frac{1+a_{1}}{2\sqrt{3}}\cdot\sqrt{\varepsilon}\int_{\Omega}u_{\varepsilon}+\frac{A}{2\sqrt{3}}\cdot\sqrt{\varepsilon}\int_{\Omega}v_{\varepsilon}$

for all $t>0$ and $\varepsilon\in(0,1)$, and thus end up with (8.6). $\Box$ In
view of the eventual bounds derived in Section 6, for small values of
$\chi_{1}$ and $\chi_{2}$ but arbitrarily large $u_{0}$ and $v_{0}$ this
readily entails an inequality for ${\mathcal{E}}_{1,\varepsilon}$ which in the
formal limit $\varepsilon\searrow 0$ indeed predicts eventual decrease
thereof.

###### Lemma 8.3

Let $n_{1}=n_{2}=2$, and for $i\in\\{1,2\\}$ let
$D_{i}>0,a_{i}>0,\lambda_{i}>0$ and $\chi_{i}>0$ be such that (8.3) holds.
Then with $\chi_{\star}>0$ taken from Lemma 6.2, one can find
$\chi_{\star\star}\in(0,\chi_{\star})$ and $C>0$ such that if
$\chi_{1}\in(0,\chi_{\star\star})$ and $\chi_{2}\in(0,\chi_{\star\star})$, and
if (1.5) holds, there exist $T_{0}=T_{0}(\chi_{1},\chi_{2},u_{0},v_{0})>0$ and
$\varepsilon_{0}=\varepsilon_{0}(\chi_{1},\chi_{2},u_{0},v_{0})\in(0,1)$ such
that for all $\varepsilon\in(0,\varepsilon_{0})$, the function
${\mathcal{E}}_{1,\varepsilon}$ defined in (8.5) satisfies

$\frac{d}{dt}{\mathcal{E}}_{1,\varepsilon}(t)+\frac{1}{C}{\mathcal{D}}_{1,\varepsilon}(t)\leq
C\sqrt{\varepsilon}\qquad\mbox{for all }t>T_{0},$ (8.9)

where for $\varepsilon\in(0,1)$,

$\displaystyle{\mathcal{D}}_{1,\varepsilon}(t)$ $\displaystyle:=$
$\displaystyle\int_{\Omega}\frac{u_{\varepsilon
x}^{2}(\cdot,t)}{u_{\varepsilon}^{2}(\cdot,t)}+\int_{\Omega}\frac{v_{\varepsilon
x}^{2}(\cdot,t)}{v_{\varepsilon}^{2}(\cdot,t)}+\int_{\Omega}(u_{\varepsilon}(\cdot,t)-u_{\star})^{2}+\int_{\Omega}(v_{\varepsilon}(\cdot,t)-v_{\star})^{2}$
(8.10) $\displaystyle+$
$\displaystyle\varepsilon^{\frac{\alpha+2}{2}}\int_{\Omega}u_{\varepsilon}^{-\alpha-4}(\cdot,t)u_{\varepsilon
x}^{2}(\cdot,t)+\varepsilon^{\frac{\alpha+2}{2}}\int_{\Omega}v_{\varepsilon}^{-\alpha-4}(\cdot,t)v_{\varepsilon
x}^{2}(\cdot,t),\qquad t>0,$

and where $u_{\star}>0$ and $v_{\star}>0$ are taken from (1.11).

Proof. We first invoke Lemma 6.2 to find $c_{1}>0$ such that if
$\chi_{1}\in(0,\chi_{\star})$ and $\chi_{2}\in(0,\chi_{\star})$, then whenever
(1.5) holds, we can find $T_{0}=T_{0}(\chi_{1},\chi_{2},u_{0},v_{0})>0$ and
$\varepsilon_{0}=\varepsilon_{0}(\chi_{1},\chi_{2},u_{0},v_{0})\in(0,1)$ such
that

$\|u_{\varepsilon}(\cdot,t)\|_{L^{\infty}(\Omega)}+\|v_{\varepsilon}(\cdot,t)\|_{L^{\infty}(\Omega)}\leq
c_{1}\qquad\mbox{for all $t>T_{0}$ and any
}\varepsilon\in(0,\varepsilon_{0}).$ (8.11)

Choosing $\chi_{\star\star}\in(0,\chi_{\star})$ small enough such that with
$A>0$ given by (8.4) we have

$\displaystyle\frac{A\chi_{\star\star}^{2}v_{\star}}{2D_{2}}\cdot
c_{1}^{2}\leq\frac{D_{1}u_{\star}}{4}\qquad\mbox{and}\qquad\frac{\chi_{\star\star}^{2}u_{\star}}{2D_{1}}\cdot
c_{1}^{2}\leq\frac{AD_{2}v_{\star}}{4},$

from Lemma 8.2 we infer that if $\chi_{1}\in(0,\chi_{\star\star})$,
$\chi_{2}\in(0,\chi_{\star\star})$ and (1.5) holds, then with $T_{0}$ and
$\varepsilon_{0}$ as above and for all $\varepsilon\in(0,\varepsilon_{0})$,

$\displaystyle\frac{d}{dt}{\mathcal{E}}_{1,\varepsilon}(t)$ $\displaystyle+$
$\displaystyle\frac{D_{1}u_{\star}}{4}\int_{\Omega}\frac{u_{\varepsilon
x}^{2}}{u_{\varepsilon}^{2}}+\frac{AD_{2}v_{\star}}{4}\int_{\Omega}\frac{v_{\varepsilon
x}^{2}}{v_{\varepsilon}^{2}}+\int_{\Omega}(u_{\varepsilon}-u_{\star})^{2}+A\int_{\Omega}(v_{\varepsilon}-v_{\star})^{2}$
$\displaystyle+$ $\displaystyle
u_{\star}\varepsilon^{\frac{\alpha+2}{2}}\int_{\Omega}u_{\varepsilon}^{-\alpha-4}u_{\varepsilon
x}^{2}+Av_{\star}\varepsilon^{\frac{\alpha+2}{2}}\int_{\Omega}v_{\varepsilon}^{-\alpha-4}v_{\varepsilon
x}^{2}$ $\displaystyle\leq$
$\displaystyle\Big{\\{}\frac{(1+a_{1})c_{1}|\Omega|}{2\sqrt{3}}+\frac{Ac_{1}|\Omega|}{2\sqrt{3}}\Big{\\}}\cdot\sqrt{\varepsilon}\qquad\mbox{for
all }t>T_{0},$

which directly yields (8.9). $\Box$ Now once more thanks to the interpolation
properties from Lemma 5.1, the dissipation rate in (8.9) dominates a certain
superlinear power of ${\mathcal{E}}_{1,\varepsilon}$ at least within bounded
time intervals:

###### Lemma 8.4

Let $n_{1}=n_{2}=2$, and let $D_{i}>0,a_{i}>0,\lambda_{i}>0$ and $\chi_{i}>0$
for $i\in\\{1,2\\}$, assuming (8.3). Then whenever (1.5) holds, for all $T>0$
there exists $C(T)>0$ such that for all $\varepsilon\in(0,1)$, with
${\mathcal{E}}_{1,\varepsilon}$ and ${\mathcal{D}}_{1,\varepsilon}$ taken from
(8.5) and (8.10) we have

${\mathcal{E}}_{1,\varepsilon}^{\frac{\alpha+2}{2}}(t)\leq
C(T){\mathcal{D}}_{1,\varepsilon}(t)+C(T)\qquad\mbox{for all $t\in(0,T)$.}$
(8.12)

Proof. According to Lemma 5.3, given $T>0$ we can find $c_{1}(T)\in(0,1)$ such
that for all $\varepsilon\in(0,1)$,

$\int_{\Omega}u_{\varepsilon}(\cdot,t)\geq
c_{1}(T)\quad\mbox{and}\quad\int_{\Omega}v_{\varepsilon}(\cdot,t)\geq
c_{1}(T)\qquad\mbox{for all }t\in(0,T),$ (8.13)

whereas Lemma 2.3 provides $c_{2}>0$ such that

$\int_{\Omega}u_{\varepsilon}(\cdot,t)+\int_{\Omega}v_{\varepsilon}(\cdot,t)\leq
c_{2}\qquad\mbox{for all }t>0$ (8.14)

whenever $\varepsilon\in(0,1)$. We now recall the definition (8.5) of
${\mathcal{E}}_{1,\varepsilon}$ to firstly estimate

$\displaystyle{\mathcal{E}}_{1,\varepsilon}(t)$ $\displaystyle\leq$
$\displaystyle\int_{\Omega}u_{\varepsilon}-u_{\star}\int_{\Omega}\ln
u_{\varepsilon}+u_{\star}\ln
u_{\star}\cdot|\Omega|+\frac{u_{\star}\varepsilon}{6}\int_{\Omega}\frac{1}{u_{\varepsilon}^{2}}$
$\displaystyle+A\int_{\Omega}v_{\varepsilon}-Av_{\star}\int_{\Omega}\ln
v_{\varepsilon}+Av_{\star}\ln
v_{\star}\cdot|\Omega|+\frac{Av_{\star}\varepsilon}{6}\int_{\Omega}\frac{1}{v_{\varepsilon}^{2}}\qquad\mbox{for
all }t>0,$

from which by nonnegativity of ${\mathcal{E}}_{1,\varepsilon}$, as asserted by
Lemma 8.1, for all $t>0$ it follows that due to (8.14),

$\displaystyle{\mathcal{E}}_{1,\varepsilon}^{\frac{\alpha+2}{2}}(t)$
$\displaystyle\leq$ $\displaystyle
8^{\frac{\alpha+2}{2}}\cdot\Bigg{\\{}u_{\star}^{\frac{\alpha+2}{2}}\cdot\bigg{\\{}-\int_{\Omega}\ln
u_{\varepsilon}\bigg{\\}}_{+}^{\frac{\alpha+2}{2}}+\Big{(}\frac{u_{\star}}{6}\Big{)}^{\frac{\alpha+2}{2}}\varepsilon^{\frac{\alpha+2}{2}}\cdot\bigg{\\{}\int_{\Omega}\frac{1}{u_{\varepsilon}^{2}}\bigg{\\}}^{\frac{\alpha+2}{2}}$
(8.15) $\displaystyle\hskip
56.9055pt+(Av_{\star})^{\frac{\alpha+2}{2}}\cdot\bigg{\\{}-\int_{\Omega}\ln
v_{\varepsilon}\bigg{\\}}_{+}^{\frac{\alpha+2}{2}}+\Big{(}\frac{Av_{\star}}{6}\Big{)}^{\frac{\alpha+2}{2}}\varepsilon^{\frac{\alpha+2}{2}}\cdot\bigg{\\{}\int_{\Omega}\frac{1}{v_{\varepsilon}^{2}}\bigg{\\}}^{\frac{\alpha+2}{2}}+c_{3}\Bigg{\\}}$

with $c_{3}:=c_{2}^{\frac{\alpha+2}{2}}+|u_{\star}\ln
u_{\star}\cdot|\Omega||^{\frac{\alpha+2}{2}}+(Ac_{2})^{\frac{\alpha+2}{2}}+|Av_{\star}\ln
v_{\star}\cdot|\Omega||^{\frac{\alpha+2}{2}}$. Here we use Lemma 5.1 ii) along
with (8.13) and Young’s inequality to see that writing
$c_{4}(T):=|\Omega|\cdot\ln\frac{1}{c_{1}(T)}+|\Omega|\cdot|\ln|\Omega||$ we
have

$\displaystyle\bigg{\\{}-\int_{\Omega}\ln
u_{\varepsilon}\bigg{\\}}_{+}^{\frac{\alpha+2}{2}}$ $\displaystyle\leq$
$\displaystyle\Bigg{\\{}|\Omega|^{\frac{3}{2}}\cdot\bigg{\\{}\int_{\Omega}\frac{u_{\varepsilon
x}^{2}}{u_{\varepsilon}^{2}}\bigg{\\}}^{\frac{1}{2}}-|\Omega|\ln\bigg{\\{}\int_{\Omega}u_{\varepsilon}\bigg{\\}}+|\Omega|\ln|\Omega|\Bigg{\\}}^{\frac{\alpha+2}{2}}$
(8.16) $\displaystyle\leq$
$\displaystyle\Bigg{\\{}|\Omega|^{\frac{3}{2}}\cdot\bigg{\\{}\int_{\Omega}\frac{u_{\varepsilon
x}^{2}}{u_{\varepsilon}^{2}}\bigg{\\}}^{\frac{1}{2}}+c_{4}\Bigg{\\}}^{\frac{\alpha+2}{2}}$
$\displaystyle\leq$
$\displaystyle\Big{(}2|\Omega|^{\frac{3}{2}}\Big{)}^{\frac{\alpha+2}{2}}\cdot\bigg{\\{}\int_{\Omega}\frac{u_{\varepsilon
x}^{2}}{u_{\varepsilon}^{2}}\bigg{\\}}^{\frac{\alpha+2}{4}}+(2c_{4})^{\frac{\alpha+2}{2}}$
$\displaystyle\leq$
$\displaystyle\Big{(}2|\Omega|^{\frac{3}{2}}\Big{)}^{\frac{\alpha+2}{2}}\cdot\int_{\Omega}\frac{u_{\varepsilon
x}^{2}}{u_{\varepsilon}^{2}}+\Big{(}2|\Omega|^{\frac{3}{2}}\Big{)}^{\frac{\alpha+2}{2}}+(2c_{4})^{\frac{\alpha+2}{2}}\qquad\mbox{for
all }t\in(0,T).$

Moreover, applying Lemma 5.1 i) to $p=2$ and $q=\alpha+2$ shows that again
thanks to (8.13) and Young’s inequality,

$\displaystyle\varepsilon^{\frac{\alpha+2}{2}}\cdot\bigg{\\{}\int_{\Omega}\frac{1}{u_{\varepsilon}^{2}}\bigg{\\}}^{\frac{\alpha+2}{2}}$
$\displaystyle\leq$
$\displaystyle\varepsilon^{\frac{\alpha+2}{2}}\cdot\Bigg{\\{}(\alpha+2)^{\frac{4}{\alpha+2}}|\Omega|^{\frac{\alpha+4}{\alpha+2}}\cdot\bigg{\\{}\int_{\Omega}u_{\varepsilon}^{-\alpha-4}u_{\varepsilon
x}^{2}\bigg{\\}}^{\frac{2}{\alpha+2}}$ $\displaystyle\hskip
56.9055pt+2^{\frac{4}{\alpha+2}}|\Omega|^{3}\cdot\bigg{\\{}\int_{\Omega}u_{\varepsilon}\bigg{\\}}^{-2}\Bigg{\\}}^{\frac{\alpha+2}{2}}$
$\displaystyle\leq$ $\displaystyle
2^{\frac{\alpha}{2}}(\alpha+2)^{2}|\Omega|^{\frac{\alpha+4}{2}}\cdot\varepsilon^{\frac{\alpha+2}{2}}\int_{\Omega}u_{\varepsilon}^{-\alpha-4}u_{\varepsilon
x}^{2}+2^{\frac{\alpha+4}{2}}|\Omega|^{\frac{3(\alpha+2)}{2}}c_{1}^{-\alpha-2}$

for all $t\in(0,T)$. Together with a similar argument for the respective
expressions containing the second solution component, this enables us to infer
from (8.15) the existence of $c_{5}>0$ and $c_{6}(T)>0$ such that for all
$\varepsilon\in(0,1)$,

$\displaystyle{\mathcal{E}}_{1,\varepsilon}^{\frac{\alpha+2}{2}}(t)\leq
c_{5}\cdot\bigg{\\{}\int_{\Omega}\frac{u_{\varepsilon
x}^{2}}{u_{\varepsilon}^{2}}+\int_{\Omega}\frac{v_{\varepsilon
x}^{2}}{v_{\varepsilon}^{2}}+\varepsilon^{\frac{\alpha+2}{2}}\int_{\Omega}u_{\varepsilon}^{-\alpha-4}u_{\varepsilon
x}^{2}+\varepsilon^{\frac{\alpha+2}{2}}\int_{\Omega}v_{\varepsilon}^{-\alpha-4}v_{\varepsilon
x}^{2}\bigg{\\}}+c_{6}(T)\quad\mbox{for all }t\in(0,T).$

In view of the definition (8.10) of ${\mathcal{D}}_{1,\varepsilon}$, this
implies (8.12). $\Box$ For drawing conclusions from the resulting autonomous
ODI for ${\mathcal{E}}_{1,\varepsilon}$, let us a quantitative consequence of
a simple comparison argument.

###### Lemma 8.5

Let $t_{0}\in\mathbb{R},T>t_{0},a>0,b>0$ and $\beta>1$, and suppose that $y\in
C^{0}([t_{0},T))\cap C^{1}((t_{0},T))$ is nonnegative and such that

$y^{\prime}(t)+ay^{\beta}(t)\leq b\qquad\mbox{for all }t\in(t_{0},T).$ (8.17)

Then

$y(t)\leq\Big{\\{}(\beta-1)a(t-t_{0})\Big{\\}}^{-\frac{1}{\beta-1}}+\Big{(}\frac{b}{a}\Big{)}^{\frac{1}{\beta}}\qquad\mbox{for
all }t\in(t_{0},T).$ (8.18)

Proof. Without loss of generality we assume that $t_{0}=0$, and observe that
$\overline{y}(t):=((\beta-1)at)^{-\frac{1}{\beta-1}}+(\frac{b}{a})^{\frac{1}{\beta}}$,
$t>0$, satisfies

$\displaystyle\overline{y}^{\prime}(t)+a\overline{y}^{\beta}(t)-b$
$\displaystyle=$
$\displaystyle(\beta-1)^{-\frac{1}{\beta-1}}a^{-\frac{1}{\beta-1}}\cdot\Big{(}-\frac{1}{\beta-1}t^{-\frac{1}{\beta-1}-1}\Big{)}+a\cdot\Big{\\{}\Big{(}(\beta-1)at\Big{)}^{-\frac{1}{\beta-1}}+\Big{(}\frac{b}{a}\Big{)}^{\frac{1}{\beta}}\Big{\\}}^{\beta}-b$
$\displaystyle\geq$
$\displaystyle-(\beta-1)^{-\frac{\beta}{\beta-1}}a^{-\frac{1}{\beta-1}}t^{-\frac{\beta}{\beta-1}}+a\cdot\Big{(}(\beta-1)at\Big{)}^{-\frac{\beta}{\beta-1}}+a\cdot\frac{b}{a}-b=0$

for all $t>0$ due to the fact that
$(\xi+\eta)^{\beta}\geq\xi^{\beta}+\eta^{\beta}$ for all $\xi\geq 0$ and
$\eta\geq 0$. Since $\overline{y}(t)\nearrow+\infty$ as $t\searrow 0$, by
continuity of $y$ at $t=0$ this readily implies the claimed inequality by
means of a comparison argument. $\Box$ We can thus reap the fruit of Lemma 8.3
and Lemma 8.4 and thereby obtain, through the dissipation mechanism expressed
in (8.9), the following preliminary decay information for our solution to the
original problem.

###### Lemma 8.6

Let $n_{1}=n_{2}=2$, let $D_{i}>0,a_{i}>0,\lambda_{i}>0$ and $\chi_{i}>0$ for
$i\in\\{1,2\\}$ be such that (8.3) holds, and let
$\chi_{1}\in(0,\chi_{\star\star})$ and $\chi_{2}\in(0,\chi_{\star\star})$ with
$\chi_{\star\star}>0$ as in Lemma 8.3. Then assuming (1.5), one can find
$T_{0}=T_{0}(\chi_{1},\chi_{2},u_{0},v_{0})>0$ such that for the limit
functions $u$ and $v$ obtained in Lemma 4.5 we have

$\int_{T_{0}}^{\infty}\int_{\Omega}u_{x}^{2}+\int_{T_{0}}^{\infty}\int_{\Omega}v_{x}^{2}+\int_{T_{0}}^{\infty}\int_{\Omega}(u-u_{\star})^{2}+\int_{T_{0}}^{\infty}\int_{\Omega}(v-v_{\star})^{2}<\infty$
(8.19)

with $u_{\star}>0$ and $v_{\star}>0$ taken from (1.11).

Proof. According to Lemma 8.3 and Lemma 6.2, supposing (1.5) to be valid we
can pick
$\varepsilon_{0}=\varepsilon_{0}(\chi_{1},\chi_{2},u_{0},v_{0})\in(0,1)$,
$T_{1}=T_{1}(\chi_{1},\chi_{2},u_{0},v_{0})>0$ and
$c_{i}=c_{i}(\chi_{1},\chi_{2},u_{0},v_{0})>0$, $i\in\\{1,2,3\\}$, such that

$\frac{d}{dt}{\mathcal{E}}_{1,\varepsilon}(t)+c_{1}{\mathcal{D}}_{1,\varepsilon}(t)\leq
c_{2}\sqrt{\varepsilon}\qquad\mbox{for all }t>T_{1}$ (8.20)

as well as

$\|u_{\varepsilon}(\cdot,t)\|_{L^{\infty}(\Omega)}+\|v_{\varepsilon}(\cdot,t)\|_{L^{\infty}(\Omega)}\leq
c_{3}\qquad\mbox{for all $t>T_{1}$ and any
}\varepsilon\in(0,\varepsilon_{0}),$ (8.21)

where ${\mathcal{E}}_{1,\varepsilon}$ and ${\mathcal{D}}_{1,\varepsilon}$ are
as defined in (8.5) and (8.10), respectively. Keeping this value of $T_{1}$
fixed, we thereafter invoke Lemma 8.4 to pick $c_{4}>0$ such that for any
choice of $\varepsilon\in(0,1)$,

$\displaystyle{\mathcal{E}}_{1,\varepsilon}^{\frac{\alpha+2}{2}}(t)\leq
c_{4}{\mathcal{D}}_{1,\varepsilon}(t)+c_{4}\qquad\mbox{for all
}t\in(0,T_{1}+2),$

which when combined with (8.20) shows that

$\displaystyle\frac{d}{dt}{\mathcal{E}}_{1,\varepsilon}(t)+\frac{c_{1}}{c_{4}}{\mathcal{E}}_{1,\varepsilon}^{\frac{\alpha+2}{2}}(t)\leq
c_{1}+c_{2}\sqrt{\varepsilon}\leq c_{1}+c_{2}\qquad\mbox{for all
$t\in(T_{1},T_{1}+2)$ and all }\varepsilon\in(0,\varepsilon_{0}).$

Through Lemma 8.5, the latter implies that

$\displaystyle{\mathcal{E}}_{1,\varepsilon}(t)\leq\Big{\\{}\frac{\alpha}{2}\cdot\frac{c_{1}}{c_{4}}\cdot(t-T_{1})\Big{\\}}^{-\frac{2}{\alpha}}+\bigg{(}\frac{c_{1}+c_{2}}{\frac{c_{1}}{c_{4}}}\bigg{)}^{\frac{2}{\alpha+2}}\qquad\mbox{for
all $t\in(T_{1},T_{1}+2)$ and }\varepsilon\in(0,\varepsilon_{0})$

and that hence, in particular,

$\displaystyle{\mathcal{E}}_{1,\varepsilon}(T_{1}+1)\leq
c_{5}:=\Big{(}\frac{2c_{4}}{c_{1}\alpha}\Big{)}^{\frac{2}{\alpha}}+\Big{(}\frac{(c_{1}+c_{2})c_{4}}{c_{1}}\Big{)}^{\frac{2}{\alpha+2}}\qquad\mbox{for
all }\varepsilon\in(0,\varepsilon_{0}).$

Using this as $\varepsilon$-independent information at the initial time
$T_{1}+1$, we now return to (8.20) to infer upon an integration therein that

$\displaystyle c_{1}\int_{T_{1}+1}^{T}{\mathcal{D}}_{1,\varepsilon}(t)dt$
$\displaystyle\leq$
$\displaystyle{\mathcal{E}}_{1,\varepsilon}(T_{1}+1)-{\mathcal{E}}_{1,\varepsilon}(T)+c_{2}\sqrt{\varepsilon}\cdot(T-T_{1}-1)$
(8.22) $\displaystyle\leq$ $\displaystyle
c_{5}+c_{2}\sqrt{\varepsilon}\cdot(T-T_{1}-1)\qquad\mbox{for all $T>T_{1}+1$
and each }\varepsilon\in(0,\varepsilon_{0}),$

again because ${\mathcal{E}}_{1,\varepsilon}$ is nonnegative. Since (8.21)
ensures that according to (8.10) we can estimate

$\displaystyle{\mathcal{D}}_{1,\varepsilon}(t)\geq\frac{1}{c_{3}^{2}}\int_{\Omega}u_{\varepsilon
x}^{2}+\frac{1}{c_{3}^{2}}\int_{\Omega}v_{\varepsilon
x}^{2}+\int_{\Omega}(u_{\varepsilon}-u_{\star})^{2}+\int_{\Omega}(v_{\varepsilon}-v_{\star})^{2}\qquad\mbox{for
all $t>T_{1}$ and }\varepsilon\in(0,\varepsilon_{0}),$

on taking $\varepsilon=\varepsilon_{j}\searrow 0$ with
$(\varepsilon_{j})_{j\in\mathbb{N}}\subset(0,1)$ as in Lemma 4.5, from (8.22)
and an argument based on lower semicontinuity of norms with respect to weak
convergence in Hilbert spaces we conclude that

$\displaystyle\frac{1}{c_{3}^{2}}\int_{T_{1}+1}^{T}\int_{\Omega}u_{x}^{2}+\frac{1}{c_{3}^{2}}\int_{T_{1}+1}^{T}\int_{\Omega}v_{x}^{2}+\int_{T_{1}+1}^{T}\int_{\Omega}(u-u_{\star})^{2}+\int_{T_{1}+1}^{T}\int_{\Omega}(v-v_{\star})^{2}\leq\frac{c_{5}}{c_{1}}\qquad\mbox{for
all }t>T_{1}+1,$

and that thus (8.19) holds with $T_{0}:=T_{1}+1$. $\Box$ We next make use of
favorable smallness properties, as implied by (8.19) for certain arbitrarily
large times which we use as new starting instants, to see on going back to
(8.9), but this time simply neglecting the positive summand
${\mathcal{D}}_{1,\varepsilon}$ therein, that $u$ and $v$ in fact approach
their expected limits in a sense much stronger than indicated in Lemma 8.6,
albeit not yet identified as spatially uniform but rather in a topology
associated with the entropy functional in (1.12).

###### Lemma 8.7

Let $n_{1}=n_{2}=2$, let $D_{i}>0,a_{i}>0,\lambda_{i}>0$ and $\chi_{i}>0$,
$i\in\\{1,2\\}$, satisfy (8.3), and let $\chi_{1}\in(0,\chi_{\star\star})$ and
$\chi_{2}\in(0,\chi_{\star\star})$ with $\chi_{\star\star}>0$ taken from Lemma
8.3. Then assuming (1.5) and letting $u,v$ and $N$ be as given by Lemma 4.5,
with $\phi_{u_{\star}}$ and $\phi_{v_{\star}}$ taken from (8.1) and (1.11) we
have

$\int_{\Omega}\phi_{u_{\star}}(u(\cdot,t))\to
0\quad\mbox{and}\quad\int_{\Omega}\phi_{v_{\star}}(v(\cdot,t))\to
0\qquad\mbox{as }(0,\infty)\setminus N\ni t\to\infty.$ (8.23)

Proof. In order to prepare our convergence argument, let us first once more
resort to Lemma 8.3 in choosing $c_{1}>0$, $T_{0}=T_{0}(u_{0},v_{0})>0$ and
$\varepsilon_{\star}=\varepsilon_{\star}(u_{0},v_{0})\in(0,1)$ such that for
all $\varepsilon\in(0,\varepsilon_{\star})$,

$\frac{d}{dt}{\mathcal{E}}_{1,\varepsilon}(t)\leq
c_{1}\sqrt{\varepsilon}\qquad\mbox{for all }t>T_{0}.$ (8.24)

Then abbreviating $c_{2}:=|\Omega|^{\frac{1}{2}}+|\Omega|^{-\frac{1}{2}}$,
given $\eta>0$ we fix $\delta>0$ conveniently small such that

$2c_{2}\sqrt{\delta}\leq\frac{u_{\star}}{2}\qquad\mbox{and}\qquad
2c_{2}\sqrt{\delta}\leq\frac{v_{\star}}{2}$ (8.25)

as well as

$\delta\leq\frac{u_{\star}\eta}{64c_{2}^{2}|\Omega|}\qquad\mbox{and}\qquad
A\delta\leq\frac{v_{\star}\eta}{64c_{2}^{2}|\Omega|}.$ (8.26)

According to Lemma 8.6, we can thereafter pick some $T_{1}>T_{0}+1$ suitably
large such that

$\int_{T_{1}-1}^{\infty}\int_{\Omega}u_{x}^{2}+\int_{T_{1}-1}^{\infty}\int_{\Omega}v_{x}^{2}+\int_{T_{1}-1}^{\infty}\int_{\Omega}(u-u_{\star})^{2}+\int_{T_{1}-1}^{\infty}\int_{\Omega}(v-v_{\star})^{2}\leq\delta,$
(8.27)

and we claim that then

$\int_{\Omega}\phi_{u_{\star}}(u(\cdot,t))+A\int_{\Omega}\phi_{v_{\star}}(v(\cdot,t))\leq\eta\qquad\mbox{for
all $t>T_{1}$ such that }t\not\in N.$ (8.28)

To verify this, given any such $t$ we first observe that due to (8.27),

$\displaystyle\int_{t-1}^{t}\bigg{\\{}\int_{\Omega}u_{x}^{2}+\int_{\Omega}v_{x}^{2}+\int_{\Omega}(u-u_{\star})^{2}+\int_{\Omega}(v-v_{\star})^{2}\bigg{\\}}\leq\delta,$

whence it is possible to find $t_{\star}=t_{\star}(t)\in(t-1,t)\setminus N$
such that

$\int_{\Omega}u_{x}^{2}(\cdot,t_{\star})+\int_{\Omega}v_{x}^{2}(\cdot,t_{\star})+\int_{\Omega}(u(\cdot,t_{\star})-u_{\star})^{2}+\int_{\Omega}(v(\cdot,t_{\star})-v_{\star})^{2}\leq\delta.$
(8.29)

In particular, this entails the existence of $x_{\star}\in\overline{\Omega}$
such that $(u(x_{\star},t_{\star})-u_{\star})^{2}\leq\frac{\delta}{|\Omega|}$
and that hence

$\displaystyle|u(x,t_{\star})-u_{\star}|$ $\displaystyle\leq$
$\displaystyle|u(x,t_{\star})-u(x_{\star},t_{\star})|+|u(x_{\star},t_{\star})-u_{\star}|$
$\displaystyle=$
$\displaystyle\bigg{|}\int_{x_{\star}}^{x}u_{x}(y,t_{\star})dy\bigg{|}+|u(x_{\star},t_{\star})-u_{\star}|$
$\displaystyle\leq$
$\displaystyle|\Omega|^{\frac{1}{2}}\cdot\bigg{\\{}\int_{\Omega}u_{x}^{2}(y,t_{\star})dy\bigg{\\}}^{\frac{1}{2}}+|u(x_{\star},t_{\star})-u_{\star}|$
$\displaystyle\leq$
$\displaystyle|\Omega|^{\frac{1}{2}}\delta^{\frac{1}{2}}+\Big{(}\frac{\delta}{|\Omega|}\Big{)}^{\frac{1}{2}}\qquad\mbox{for
all }x\in\Omega,$

which along with an identical argument for $v$ shows that

$\|u(\cdot,t_{\star})-u_{\star}\|_{L^{\infty}(\Omega)}\leq
c_{2}\sqrt{\delta}\qquad\mbox{and}\qquad\|v(\cdot,t_{\star})-v_{\star}\|_{L^{\infty}(\Omega)}\leq
c_{2}\sqrt{\delta}.$ (8.30)

Now since $t_{\star}\in(0,\infty)\setminus N$, Corollary 6.3 applies so as to
warrant that with some
$\varepsilon_{\star\star}=\varepsilon_{\star\star}(t_{\star})\in(0,\varepsilon_{\star})$
and $(\varepsilon_{j})_{j\in\mathbb{N}}\subset(0,1)$ as in Lemma 4.5 we have

$\displaystyle\|u_{\varepsilon}(\cdot,t_{\star})-u(\cdot,t_{\star})\|_{L^{\infty}(\Omega)}+\|v_{\varepsilon}(\cdot,t_{\star})-v(\cdot,t_{\star})\|_{L^{\infty}(\Omega)}\leq
c_{2}\sqrt{\delta}\qquad\mbox{for all
}\varepsilon\in(\varepsilon_{j})_{j\in\mathbb{N}}\cap(0,\varepsilon_{\star\star}),$

together with (8.30) ensuring that

$\|u_{\varepsilon}(\cdot,t_{\star})-u_{\star}\|_{L^{\infty}(\Omega)}\leq
2c_{2}\sqrt{\delta}\quad\mbox{and}\quad\|v_{\varepsilon}(\cdot,t_{\star})-v_{\star}\|_{L^{\infty}(\Omega)}\leq
2c_{2}\sqrt{\delta}\qquad\mbox{for all
}\varepsilon\in(\varepsilon_{j})_{j\in\mathbb{N}}\cap(0,\varepsilon_{\star\star}).$
(8.31)

According to our requirements on $\delta$ in (8.25), these estimates
especially entail the inequalities

$u_{\varepsilon}(\cdot,t_{\star})\geq
u_{\star}-2c_{2}\sqrt{\delta}\geq\frac{u_{\star}}{2}\quad\mbox{and}\quad
v_{\varepsilon}(\cdot,t_{\star})\geq
v_{\star}-2c_{2}\sqrt{\delta}\geq\frac{v_{\star}}{2}\quad\mbox{in $\Omega$
\qquad for all
}\varepsilon\in(\varepsilon_{j})_{j\in\mathbb{N}}\cap(0,\varepsilon_{\star\star}),$
(8.32)

which firstly enables us to conclude from (8.2) in conjunction with (8.31) and
(8.26) that for all
$\varepsilon\in(\varepsilon_{j})_{j\in\mathbb{N}}\cap(0,\varepsilon_{\star\star})$,

$\displaystyle\int_{\Omega}\phi_{u_{\star}}(u_{\varepsilon}(\cdot,t_{\star}))\leq\frac{2}{u_{\star}}\int_{\Omega}\Big{(}u_{\varepsilon}(\cdot,t_{\star})-u_{\star}\Big{)}^{2}\leq\frac{2|\Omega|}{u_{\star}}\cdot\|u_{\varepsilon}(\cdot,t_{\star})-u_{\star}\|_{L^{\infty}(\Omega)}^{2}\leq\frac{2|\Omega|}{u_{\star}}\cdot(2c_{2}\sqrt{\delta})^{2}\leq\frac{\eta}{8}$
(8.33)

and similarly

$\displaystyle
A\int_{\Omega}\phi_{v_{\star}}(v_{\varepsilon}(\cdot,t_{\star}))\leq\frac{\eta}{8}.$
(8.34)

Secondly, (8.32) guarantees that if we pick
$\varepsilon_{\star\star\star}=\varepsilon_{\star\star\star}(t_{\star})\in(0,\varepsilon_{\star\star})$
small enough such that

$\displaystyle\frac{2|\Omega|\varepsilon_{\star\star\star}}{3u_{\star}}\leq\frac{\eta}{8}\qquad\mbox{and}\qquad\frac{2A|\Omega|\varepsilon_{\star\star\star}}{3v_{\star}}\leq\frac{\eta}{8},$

then in the contributions to ${\mathcal{E}}_{1,\varepsilon}$ containing the
factor $\varepsilon$ we can estimate

$\displaystyle\frac{u_{\star}\varepsilon}{6}\int_{\Omega}\frac{1}{u_{\varepsilon}^{2}(\cdot,t_{\star})}\leq\frac{u_{\star}\varepsilon}{6}\cdot\Big{(}\frac{2}{u_{\star}}\Big{)}^{2}|\Omega|\leq\frac{\eta}{8}\qquad\mbox{for
all
}\varepsilon\in(\varepsilon_{j})_{j\in\mathbb{N}}\cap(0,\varepsilon_{\star\star\star})$

and

$\displaystyle\frac{Av_{\star}\varepsilon}{6}\int_{\Omega}\frac{1}{v_{\varepsilon}^{2}(\cdot,t_{\star})}\leq\frac{Av_{\star}\varepsilon}{6}\cdot\Big{(}\frac{2}{v_{\star}}\Big{)}^{2}|\Omega|\leq\frac{\eta}{8}\qquad\mbox{for
all
}\varepsilon\in(\varepsilon_{j})_{j\in\mathbb{N}}\cap(0,\varepsilon_{\star\star\star}).$

When combined with (8.33) and (8.34), in view of (8.5) these inequalities show
that

$\displaystyle{\mathcal{E}}_{1,\varepsilon}(t_{\star})\leq
4\cdot\frac{\eta}{8}=\frac{\eta}{2}\qquad\mbox{for all
}\varepsilon\in(\varepsilon_{j})_{j\in\mathbb{N}}\cap(0,\varepsilon_{\star\star\star}),$

so that letting
$\varepsilon_{\star\star\star\star}=\varepsilon_{\star\star\star\star}(t_{\star})\in(0,\varepsilon_{\star\star\star})$
be such that
$c_{1}\sqrt{\varepsilon_{\star\star\star\star}}\leq\frac{\eta}{2}$, on
integrating (8.24) we infer that at the time in question we have

$\displaystyle{\mathcal{E}}_{1,\varepsilon}(t)\leq{\mathcal{E}}_{1,\varepsilon}(t_{\star})+c_{1}\sqrt{\varepsilon}\cdot(t-t_{\star})\leq\frac{\eta}{2}+c_{1}\sqrt{\varepsilon}\leq\frac{\eta}{2}+\frac{\eta}{2}=\eta\qquad\mbox{for
all
}\varepsilon\in(\varepsilon_{j})_{j\in\mathbb{N}}\cap(0,\varepsilon_{\star\star\star\star}),$
(8.35)

because $t_{\star}\geq t-1$. Since from Lemma 4.5 and our assumption that
$t\not\in N$ we know that $u_{\varepsilon}(\cdot,t)\to u(\cdot,t)$ and
$v_{\varepsilon}(\cdot,t)\to v(\cdot,t)$ a.e. in $\Omega$ as
$\varepsilon=\varepsilon_{j}\searrow 0$, upon an application of Fatou’s lemma
we readily obtain (8.28), and thus the statement of the lemma, as a
consequence of (8.35). $\Box$ Now thanks to the eventual precompactness
features implied by Corollary 6.3, due to the positivity of
$\phi_{\xi_{\star}}$ outside the point $\xi_{\star}>0$ the latter readily
implies the desired convergence statement.

###### Lemma 8.8

Let $n_{1}=n_{2}=2$, let $D_{i}>0,a_{i}>0,\lambda_{i}>0$ and $\chi_{i}>0$ for
$i\in\\{1,2\\}$ be such that (8.3) is fulfilled, and let
$\chi_{1}\in(0,\chi_{\star\star})$ and $\chi_{2}\in(0,\chi_{\star\star})$ with
$\chi_{\star\star}>0$ as in Lemma 8.3. Then whenever (1.5) holds, as
$t\to\infty$, the limit functions $u$ and $v$ obtained in Lemma 4.5 satisfy

$u(\cdot,t)\to u_{\star}\quad\mbox{in
}L^{\infty}(\Omega)\qquad\mbox{and}\qquad v(\cdot,t)\to v_{\star}\quad\mbox{in
}L^{\infty}(\Omega),$ (8.36)

where $u_{\star}>0$ and $v_{\star}>0$ are as in (1.11).

Proof. From Corollary 6.3 we know that there exists $T_{0}>0$ such that
$(u(\cdot,t))_{t>T_{0}}$ is bounded in $W^{1,2}(\Omega)$ and hence relatively
compact in $C^{0}(\overline{\Omega})$, whence if (8.36) was false, the we
could find $(t_{k})_{k\in\mathbb{N}}\subset(T_{0},\infty)$ and $u_{\infty}\in
C^{0}(\overline{\Omega})$ such that $u_{\infty}\not\equiv u_{\star}$ and that
$t_{k}\to\infty$ as well as $u(\cdot,t_{k})\to u_{\infty}$ in
$L^{\infty}(\Omega)$ as $k\to\infty$. As in view of Corollary 6.3 we may
assume $u$ to be continuous in $\overline{\Omega}\times(T_{0},\infty)$, by
density of $[t_{k},t_{k}+1]\setminus N$ in $[t_{k},t_{k}+1]$ we can pick
$\widehat{t}_{k}\in[t_{k},t_{k}+1]$ such that
$\|u(\cdot,\widehat{t}_{k})-u(\cdot,t_{k})\|_{L^{\infty}(\Omega)}\leq\frac{1}{k}$,
meaning that also $u(\cdot,\widehat{t}_{k})\to u_{\infty}$ in
$L^{\infty}(\Omega)$ as $k\to\infty$. As the function $\phi_{u_{\star}}$ from
(8.1) is positive in $(0,\infty)\setminus\\{u_{\star}\\}$ by Lemma 8.1,
however, the hypothesis $u_{\infty}\not\equiv u_{\star}$ implies that
$\int_{\Omega}\phi_{u_{\star}}(u(\cdot,\widehat{t}_{k}))\not\to 0$ as
$k\to\infty$, in contradiction to Lemma 8.7. Along with a similar argument for
$v$, this establishes the claim. $\Box$ The proof of our main result on
kinetics-driven stabilization has thereby already been accomplished:
Proof of Theorem 1.2. We only need to combine Corollary 6.3 with Lemma 8.8.
$\Box$

##  9 The case $\lambda_{2}\leq a_{2}\lambda_{1}$. Proof of Theorem 1.3

In the context addressed in Theorem 1.3, in view of Lemma 7.1 the form of the
functional in (1.14) suggests to choose still $n_{1}=2$ but now rather
$n_{2}=1$. Our analysis will then quite closely follow the lines presented in
the previous section, so that here it will be sufficient to concentrate on the
main modifications only.
The fundament for convergence is constituted by a natural counterpart of Lemma
8.2.

###### Lemma 9.1

Let $n_{1}=2$ and $n_{2}=1$, and suppose that $D_{i}>0,a_{i}>0,\lambda_{i}>0$
and $\chi_{i}>0$ for $i\in\\{1,2\\}$, and that

$\lambda_{2}\leq a_{2}\lambda_{1}.$ (9.1)

Then with $A=\frac{a_{1}}{a_{2}}$ as before and $\phi_{\lambda_{1}}$ as
determined by (8.1), for

$\displaystyle{\mathcal{E}}_{2,\varepsilon}(t):=\int_{\Omega}\phi_{\lambda_{1}}(u_{\varepsilon}(\cdot,t))+\frac{\lambda_{1}\varepsilon}{6}\int_{\Omega}\frac{1}{u_{\varepsilon}^{2}(\cdot,t)}+A\int_{\Omega}v_{\varepsilon}(\cdot,t)$
$\displaystyle+$
$\displaystyle\frac{A}{2\lambda_{2}}\int_{\Omega}v_{\varepsilon}^{2}(\cdot,t)+\frac{A\varepsilon}{2\lambda_{2}}\int_{\Omega}\frac{1}{v_{\varepsilon}(\cdot,t)},$
(9.2) $\displaystyle\qquad t\geq 0,\ \varepsilon\in(0,1),$

we have

$\displaystyle\frac{d}{dt}{\mathcal{E}}_{2,\varepsilon}(t)$ $\displaystyle+$
$\displaystyle\bigg{\\{}\frac{D_{1}\lambda_{1}}{2}-\frac{A\chi_{2}^{2}}{2D_{2}\lambda_{2}}\|u_{\varepsilon}(\cdot,t)\|_{L^{\infty}(\Omega)}^{2}\|v_{\varepsilon}(\cdot,t)\|_{L^{\infty}(\Omega)}^{2}\bigg{\\}}\cdot\int_{\Omega}\frac{u_{\varepsilon
x}^{2}}{u_{\varepsilon}^{2}}$ (9.3) $\displaystyle+$
$\displaystyle\bigg{\\{}\frac{AD_{2}}{2\lambda_{2}}-\frac{\lambda_{1}^{2}\chi_{1}^{2}}{2D_{1}}\bigg{\\}}\cdot\int_{\Omega}\frac{v_{\varepsilon
x}^{2}}{v_{\varepsilon}^{2}}$ $\displaystyle+$
$\displaystyle\int_{\Omega}(u_{\varepsilon}(\cdot,t)-\lambda_{1})^{2}+\frac{A}{\lambda_{2}}\int_{\Omega}v_{\varepsilon}^{3}$
$\displaystyle+$
$\displaystyle\lambda_{1}\varepsilon^{\frac{\alpha+2}{2}}\int_{\Omega}u_{\varepsilon}^{-\alpha-4}u_{\varepsilon
x}^{2}+\frac{A}{\lambda_{2}}\varepsilon^{\frac{\alpha+2}{2}}\int_{\Omega}v_{\varepsilon}^{-\alpha-3}v_{\varepsilon
x}^{2}$ $\displaystyle\leq$
$\displaystyle\frac{1+Aa_{2}+2Aa_{2}\lambda_{2}^{-1}}{2\sqrt{3}}\cdot\sqrt{\varepsilon}\int_{\Omega}u_{\varepsilon}+\frac{A+2A\lambda_{2}^{-1}}{2\sqrt{3}}\cdot\sqrt{\varepsilon}\int_{\Omega}v_{\varepsilon}$

for all $t>0$ and $\varepsilon\in(0,1)$.

Proof. Combining Lemma 2.2 with Lemma 7.1 i) and iii), on dropping several
nonnegative summands we see that

$\displaystyle\frac{d}{dt}{\mathcal{E}}_{2,\varepsilon}(t)+D_{1}\lambda_{1}\int_{\Omega}\frac{u_{\varepsilon
x}^{2}}{u_{\varepsilon}^{2}}+\lambda_{1}\varepsilon^{\frac{\alpha+2}{2}}\int_{\Omega}u_{\varepsilon}^{-\alpha-4}u_{\varepsilon
x}^{2}$ (9.4)
$\displaystyle+\frac{AD_{2}}{\lambda_{2}}\int_{\Omega}v_{\varepsilon
x}^{2}+\frac{A}{\lambda_{2}}\varepsilon^{\frac{\alpha+2}{2}}\int_{\Omega}v_{\varepsilon}^{-\alpha-3}v_{\varepsilon
x}^{2}$ $\displaystyle\leq$
$\displaystyle\Big{(}\lambda_{1}+\frac{\sqrt{\varepsilon}}{2\sqrt{3}}\Big{)}\int_{\Omega}u_{\varepsilon}-\int_{\Omega}u_{\varepsilon}^{2}+a_{1}\int_{\Omega}u_{\varepsilon}v_{\varepsilon}$
$\displaystyle+\lambda_{1}\chi_{1}\int_{\Omega}\frac{u_{\varepsilon
x}}{u_{\varepsilon}}v_{\varepsilon
x}-\lambda_{1}^{2}|\Omega|+\lambda_{1}\int_{\Omega}u_{\varepsilon}-a_{1}\lambda_{1}\int_{\Omega}v_{\varepsilon}$
$\displaystyle+A\Big{(}\lambda_{2}+\frac{\sqrt{\varepsilon}}{2\sqrt{3}}\Big{)}\int_{\Omega}v_{\varepsilon}-A\int_{\Omega}v_{\varepsilon}^{2}-Aa_{2}\int_{\Omega}u_{\varepsilon}v_{\varepsilon}+\frac{Aa_{2}\sqrt{\varepsilon}}{2\sqrt{3}}\int_{\Omega}u_{\varepsilon}$
$\displaystyle-\frac{A\chi_{2}}{\lambda_{2}}\int_{\Omega}v_{\varepsilon}u_{\varepsilon
x}v_{\varepsilon
x}+A\int_{\Omega}v_{\varepsilon}^{2}-\frac{A}{\lambda_{2}}\int_{\Omega}v_{\varepsilon}^{3}-\frac{Aa_{2}}{\lambda_{2}}\int_{\Omega}u_{\varepsilon}v_{\varepsilon}^{2}$
$\displaystyle-\frac{A\varepsilon}{\lambda_{2}}\int_{\Omega}\frac{2v_{\varepsilon}^{2}+3v_{\varepsilon}}{6v_{\varepsilon}^{2}+2\varepsilon}(\lambda_{2}-v_{\varepsilon}-a_{2}u_{\varepsilon})$
$\displaystyle=$
$\displaystyle-\int_{\Omega}(u_{\varepsilon}-\lambda_{1})^{2}+(A\lambda_{2}-a_{1}\lambda_{1})\int_{\Omega}v_{\varepsilon}-\frac{A}{\lambda_{2}}\int_{\Omega}v_{\varepsilon}^{3}-\frac{Aa_{2}}{\lambda_{2}}\int_{\Omega}u_{\varepsilon}v_{\varepsilon}^{2}$
$\displaystyle+\lambda_{1}\chi_{1}\int_{\Omega}\frac{u_{\varepsilon
x}}{u_{\varepsilon}}v_{\varepsilon
x}-\frac{A\chi_{2}}{\lambda_{2}}\int_{\Omega}v_{\varepsilon}u_{\varepsilon
x}v_{\varepsilon x}$
$\displaystyle+\frac{(1+Aa_{2})\sqrt{\varepsilon}}{2\sqrt{3}}\int_{\Omega}u_{\varepsilon}+\frac{A\sqrt{\varepsilon}}{2\sqrt{3}}\int_{\Omega}v_{\varepsilon}$
$\displaystyle-\frac{A\varepsilon}{\lambda_{2}}\int_{\Omega}\frac{2v_{\varepsilon}^{2}+3v_{\varepsilon}}{6v_{\varepsilon}^{2}+2\varepsilon}(\lambda_{2}-v_{\varepsilon}-a_{2}u_{\varepsilon})\qquad\mbox{for
all $t>0$ and $\varepsilon\in(0,1)$.}$

Here we note that according to (9.1) we have
$A\lambda_{2}-a_{1}\lambda_{1}=\frac{a_{1}\lambda_{2}}{a_{2}}-a_{1}\lambda_{1}\leq
0$, and that by Young’s inequality,

$\displaystyle\lambda_{1}\chi_{1}\int_{\Omega}\frac{u_{\varepsilon
x}}{u_{\varepsilon}}v_{\varepsilon x}$ $\displaystyle\leq$
$\displaystyle\frac{D_{1}\lambda_{1}}{2}\int_{\Omega}\frac{u_{\varepsilon
x}^{2}}{u_{\varepsilon}^{2}}+\frac{\lambda_{1}^{2}\chi_{1}^{2}}{2D_{1}}\int_{\Omega}v_{\varepsilon
x}^{2}$

and

$\displaystyle-\frac{A\chi_{2}}{\lambda_{2}}\int_{\Omega}v_{\varepsilon}u_{\varepsilon
x}v_{\varepsilon x}$ $\displaystyle\leq$
$\displaystyle\frac{AD_{2}}{2\lambda_{2}}\int_{\Omega}v_{\varepsilon
x}^{2}+\frac{A\chi_{2}^{2}}{2D_{2}\lambda_{2}}\int_{\Omega}v_{\varepsilon}^{2}u_{\varepsilon
x}^{2}$ $\displaystyle\leq$
$\displaystyle\frac{AD_{2}}{2\lambda_{2}}\int_{\Omega}v_{\varepsilon
x}^{2}+\frac{A\chi_{2}^{2}}{2D_{2}\lambda_{2}}\|u_{\varepsilon}\|_{L^{\infty}(\Omega)}^{2}\|v_{\varepsilon}\|_{L^{\infty}(\Omega)}^{2}\int_{\Omega}\frac{u_{\varepsilon
x}^{2}}{u_{\varepsilon}^{2}}$

for all $t>0$ and $\varepsilon\in(0,1)$. As moreover maximizing
$\varphi(s):=\frac{2s^{2}+3s}{6s^{2}+2\varepsilon}$, $s\geq 0$, shows that

$\displaystyle-\frac{A\varepsilon}{\lambda_{2}}\int_{\Omega}\frac{2v_{\varepsilon}^{2}+3v_{\varepsilon}}{6v_{\varepsilon}^{2}+2\varepsilon}(\lambda_{2}-v_{\varepsilon}-a_{2}u_{\varepsilon})$
$\displaystyle\leq$
$\displaystyle\frac{A\varepsilon}{\lambda_{2}}\|\varphi\|_{L^{\infty}((0,\infty))}\int_{\Omega}(v_{\varepsilon}+a_{2}u_{\varepsilon})$
$\displaystyle=$
$\displaystyle\frac{A\varepsilon}{\lambda_{2}}\cdot\frac{1}{3}\cdot\bigg{\\{}\int_{\Omega}v_{\varepsilon}+a_{2}\int_{\Omega}u_{\varepsilon}\bigg{\\}}$
$\displaystyle<$
$\displaystyle\frac{A\lambda_{2}^{-1}}{\sqrt{3}}\cdot\sqrt{\varepsilon}\cdot\bigg{\\{}\int_{\Omega}v_{\varepsilon}+a_{2}\int_{\Omega}u_{\varepsilon}\bigg{\\}}$

for all $t>0$ and $\varepsilon\in(0,1)$, from (9.4) we directly obtain (9.3).
$\Box$ This implies an inequality of the form in Lemma 8.3.

###### Lemma 9.2

Let $n_{1}=2$ and $n_{2}=1$, let $D_{i}>0,a_{i}>0,\lambda_{i}>0$ and
$\chi_{i}>0$ for $i\in\\{1,2\\}$, and suppose that (9.1) holds. Then with
$\chi_{\star}>0$ as in Lemma 6.2, there exists
$\chi_{\star\star}\in(0,\chi_{\star})$ and $C>0$ such that if
$\chi_{1}\in(0,\chi_{\star\star}),\chi_{2}\in(0,\chi_{\star\star})$ and (1.5)
is valid, then there exist $T_{0}=T_{0}(\chi_{1},\chi_{2},u_{0},v_{0})>0$ and
$\varepsilon_{0}=\varepsilon_{0}(\chi_{1},\chi_{2},u_{0},v_{0})\in(0,1)$ such
that whenever $\varepsilon\in(0,\varepsilon_{0})$, for
${\mathcal{E}}_{2,\varepsilon}$ as in (9.2) we have

$\frac{d}{dt}{\mathcal{E}}_{2,\varepsilon}(t)+\frac{1}{C}{\mathcal{D}}_{2,\varepsilon}(t)\leq
C\sqrt{\varepsilon}\qquad\mbox{for all }t>T_{0},$ (9.5)

where

$\displaystyle{\mathcal{D}}_{2,\varepsilon}(t)$ $\displaystyle:=$
$\displaystyle\int_{\Omega}\frac{u_{\varepsilon
x}^{2}(\cdot,t)}{u_{\varepsilon}^{2}(\cdot,t)}+\int_{\Omega}v_{\varepsilon
x}^{2}(\cdot,t)+\int_{\Omega}(u_{\varepsilon}(\cdot,t)-\lambda)^{2}+\int_{\Omega}v_{\varepsilon}^{3}(\cdot,t)$
(9.6) $\displaystyle+$
$\displaystyle\varepsilon^{\frac{\alpha+2}{2}}\int_{\Omega}u_{\varepsilon}^{-\alpha-4}(\cdot,t)u_{\varepsilon
x}^{2}(\cdot,t)+\varepsilon^{\frac{\alpha+2}{2}}\int_{\Omega}v_{\varepsilon}^{-\alpha-3}(\cdot,t)v_{\varepsilon
x}^{2}(\cdot,t),\qquad t>0,$

and where $u_{\star}>0$ and $v_{\star}>0$ are taken from (1.11).

Proof. Again on the basis of Lemma 6.2, this can be derived from Lemma 9.1 in
much the same manner as Lemma 8.3 was deduced from Lemma 8.2. $\Box$ Up to
modifications in technical details, the strategy in the proof of Lemma 8.4
finds its analogue in the following.

###### Lemma 9.3

Let $n_{1}=2$ and $n_{2}=1$, let $D_{i}>0,a_{i}>0,\lambda_{i}>0$ and
$\chi_{i}>0$ for $i\in\\{1,2\\}$, and assume (9.1) as well as (1.5). Then for
all $T>0$ there exists $C(T)>0$ such that for all $\varepsilon\in(0,1)$, the
functions ${\mathcal{E}}_{2,\varepsilon}$ and ${\mathcal{D}}_{2,\varepsilon}$
defined in (9.2) and (9.6) satisfy

${\mathcal{E}}_{2,\varepsilon}^{\frac{\alpha+2}{2}}(t)\leq
C(T){\mathcal{D}}_{2,\varepsilon}(t)+C(T)\qquad\mbox{for all $t\in(0,T)$.}$
(9.7)

Proof. We proceed in a way similar to that in Lemma 8.4 and first note that

$\inf_{\varepsilon\in(0,1)}\inf_{t\in(0,T)}\int_{\Omega}u_{\varepsilon}>0,\quad\inf_{\varepsilon\in(0,1)}\inf_{t\in(0,T)}\int_{\Omega}v_{\varepsilon}>0\quad\mbox{and}\quad\sup_{\varepsilon\in(0,1)}\sup_{t\in(0,T)}\bigg{\\{}\int_{\Omega}u_{\varepsilon}+\int_{\Omega}v_{\varepsilon}\bigg{\\}}<\infty,$
(9.8)

to derive the existence of $c_{1}(T)>0$ such that for all $t\in(0,T)$ and each
$\varepsilon\in(0,1)$,

$\displaystyle{\mathcal{E}}_{2,\varepsilon}^{\frac{\alpha+2}{2}}(t)$
$\displaystyle\leq$ $\displaystyle c_{1}(T)\cdot\bigg{\\{}-\int_{\Omega}\ln
u_{\varepsilon}\bigg{\\}}_{+}^{\frac{\alpha+2}{2}}+c_{1}(T)\varepsilon^{\frac{\alpha+2}{2}}\cdot\bigg{\\{}\int_{\Omega}\frac{1}{u_{\varepsilon}^{2}}\bigg{\\}}^{\frac{\alpha+2}{2}}$
(9.9)
$\displaystyle+c_{1}(T)\cdot\bigg{\\{}\int_{\Omega}v_{\varepsilon}^{2}\bigg{\\}}^{\frac{\alpha+2}{2}}+c_{1}(T)\varepsilon^{\frac{\alpha+2}{2}}\cdot\bigg{\\{}\int_{\Omega}\frac{1}{v_{\varepsilon}}\bigg{\\}}^{\frac{\alpha+2}{2}}+c_{1}(T),$

where again combining Lemma 5.1 with (9.8) and Young’s inequality provides
$c_{2}(T)>0$ fulfilling

$\displaystyle c_{1}(T)\cdot\bigg{\\{}-\int_{\Omega}\ln
u_{\varepsilon}\bigg{\\}}_{+}^{\frac{\alpha+2}{2}}+c_{1}(T)\varepsilon^{\frac{\alpha+2}{2}}\cdot\bigg{\\{}\int_{\Omega}\frac{1}{u_{\varepsilon}^{2}}\bigg{\\}}^{\frac{\alpha+2}{2}}$
(9.10) $\displaystyle\leq$ $\displaystyle
c_{2}(T)\int_{\Omega}\frac{u_{\varepsilon
x}^{2}}{u_{\varepsilon}^{2}}+c_{2}(T)\varepsilon^{\frac{\alpha+2}{2}}\int_{\Omega}u_{\varepsilon}^{-\alpha-4}u_{\varepsilon
x}^{2}+c_{2}(T)\qquad\mbox{for all $t\in(0,T)$ and }\varepsilon\in(0,1).$

Another application of Lemma 5.1 i), now to $p=1$ and $q=\alpha+1$, reveals
that again due to (9.8) and Young’s inequality we can moreover find
$c_{3}(T)>0$ such that

$\displaystyle
c_{1}(T)\varepsilon^{\frac{\alpha+2}{2}}\cdot\bigg{\\{}\int_{\Omega}\frac{1}{v_{\varepsilon}}\bigg{\\}}^{\frac{\alpha+2}{2}}$
$\displaystyle\leq$ $\displaystyle
c_{1}(T)\varepsilon^{\frac{\alpha+2}{2}}\cdot\Bigg{\\{}(\alpha+1)^{\frac{2}{\alpha+1}}|\Omega|^{\frac{\alpha+2}{\alpha+1}}\cdot\bigg{\\{}\int_{\Omega}v_{\varepsilon}^{-\alpha-3}v_{\varepsilon
x}^{2}\bigg{\\}}^{\frac{1}{\alpha+1}}$ (9.11) $\displaystyle\hskip
68.2866pt+2^{\frac{2}{\alpha+1}}|\Omega|^{2}\cdot\bigg{\\{}\int_{\Omega}v_{\varepsilon}\bigg{\\}}^{-1}\Bigg{\\}}^{\frac{\alpha+2}{2}}$
$\displaystyle\leq$ $\displaystyle
c_{3}(T)\varepsilon^{\frac{\alpha+2}{2}}\cdot\bigg{\\{}\int_{\Omega}v_{\varepsilon}^{-\alpha-3}v_{\varepsilon
x}^{2}\bigg{\\}}^{\frac{\alpha+2}{2(\alpha+1)}}+c_{3}(T)\varepsilon^{\frac{\alpha+2}{2}}$
$\displaystyle\leq$ $\displaystyle
c_{3}(T)\varepsilon^{\frac{\alpha+2}{2}}\int_{\Omega}v_{\varepsilon}^{-\alpha-3}v_{\varepsilon
x}^{2}+2c_{3}(T)\qquad\mbox{for all $t\in(0,T)$ and }\varepsilon\in(0,1),$

because $\frac{\alpha+2}{2(\alpha+1)}\leq 1$. Finally, the Gagliardo-Nirenberg
inequality along with (9.8) and Young’s inequality ensures the existence of
$c_{4}>0$ and $c_{5}(T)>0$ such that for all $t\in(0,T)$ and any
$\varepsilon\in(0,1)$,

$\displaystyle
c_{1}(T)\cdot\bigg{\\{}\int_{\Omega}v_{\varepsilon}^{2}\bigg{\\}}^{\frac{\alpha+2}{2}}$
$\displaystyle\leq$ $\displaystyle
c_{1}(T)\cdot\bigg{\\{}c_{4}\|v_{\varepsilon
x}\|_{L^{2}(\Omega)}^{\frac{\alpha+2}{3}}\|v_{\varepsilon}\|_{L^{1}(\Omega)}^{\frac{2(\alpha+2)}{3}}+c_{4}\|v_{\varepsilon}\|_{L^{1}(\Omega)}^{\alpha+2}\bigg{\\}}$
$\displaystyle\leq$ $\displaystyle c_{5}(T)\|v_{\varepsilon
x}\|_{L^{2}(\Omega)}^{\frac{\alpha+2}{3}}+c_{5}(T)$ $\displaystyle\leq$
$\displaystyle c_{5}(T)\int_{\Omega}v_{\varepsilon x}^{2}+2c_{5}(T),$

for $\frac{\alpha+2}{3}\leq 2$. Together with (9.10), (9.11) and (9.9), this
immediately leads to (9.7). $\Box$ Therefore, Lemma 9.2 implies decay in a yet
very weak form similar to that in Lemma 8.6.

###### Lemma 9.4

Let $n_{1}=2$ and $n_{2}=1$, let $D_{i}>0,a_{i}>0,\lambda_{i}>0$ and
$\chi_{i}>0$ for $i\in\\{1,2\\}$, assume (9.1), and let
$\chi_{1}\in(0,\chi_{\star\star})$ and $\chi_{2}\in(0,\chi_{\star\star})$ with
$\chi_{\star\star}>0$ as in Lemma 9.2. Then whenever (1.5) holds, there exists
$T_{0}=T_{0}(\chi_{1},\chi_{2},u_{0},v_{0})>0$ such that $u$ and $v$ from
Lemma 4.5 satisfy

$\int_{T_{0}}^{\infty}\int_{\Omega}u_{x}^{2}+\int_{T_{0}}^{\infty}\int_{\Omega}v_{x}^{2}+\int_{T_{0}}^{\infty}\int_{\Omega}(u-\lambda_{1})^{2}+\int_{T_{0}}^{\infty}\int_{\Omega}v^{3}<\infty.$
(9.12)

Proof. Once more thanks to Lemma 6.2, this can be seen by exploiting Lemma 9.2
together with Lemma 9.3 in essentially the same way as Lemma 8.3 and Lemma 8.4
have been used in the derivation of Lemma 8.6. $\Box$ With this information
returning to the inequality from Lemma 9.2 yields stabilization in a sense
paralleling that of Lemma 8.7.

###### Lemma 9.5

Let $n_{1}=2$ and $n_{2}=1$, let $D_{i}>0,a_{i}>0,\lambda_{i}>0$ and
$\chi_{i}>0$ for $i\in\\{1,2\\}$ be such that (9.1) is valid, and let
$\chi_{1}\in(0,\chi_{\star\star})$ and $\chi_{2}\in(0,\chi_{\star\star})$ with
$\chi_{\star\star}>0$ as in Lemma 8.3. Then assuming (1.5) and letting $u,v$
and $N$ be as given by Lemma 4.5 and $\phi_{\lambda_{1}}$ be as defined
through (8.1), we have

$\int_{\Omega}\phi_{\lambda_{1}}(u(\cdot,t))\to
0\quad\mbox{and}\quad\int_{\Omega}v^{3}(\cdot,t)\to 0\qquad\mbox{as
}(0,\infty)\setminus N\ni t\to\infty.$ (9.13)

Proof. A verification of this can be achieved by adapting the proof of Lemma
8.7 in an obvious manner. $\Box$ Finally, by compactness the latter can be
turned into uniform convergence as in Lemma 8.8.

###### Lemma 9.6

Let $n_{1}=2$ and $n_{2}=1$, let $D_{i}>0,a_{i}>0,\lambda_{i}>0$ and
$\chi_{i}>0$ for $i\in\\{1,2\\}$ be such that (9.1) holds, and let
$\chi_{1}\in(0,\chi_{\star\star})$ and $\chi_{2}\in(0,\chi_{\star\star})$ with
$\chi_{\star\star}>0$ as in Lemma 9.2. Then whenever (1.5) is satisfied, the
limit functions $u$ and $v$ in Lemma 4.5 have the properties that

$\displaystyle u(\cdot,t)\to\lambda_{1}\quad\mbox{in
}L^{\infty}(\Omega)\qquad\mbox{and}\qquad v(\cdot,t)\to 0\quad\mbox{in
}L^{\infty}(\Omega)$

as $t\to\infty$.

Proof. Again relying on Corollary 6.3 and Lemma 8.1, one can readily obtain
this as a consequence of Lemma 9.5 by means of an argument in the flavor of
that presented in the proof of Lemma 8.8. $\Box$ We have thereby established
our main result on asymptotic dominance of the predator population when
$\lambda_{2}\leq a_{2}\lambda_{1}$.
Proof of Theorem 1.3. All statements have been verified in Corollary 6.3 and
Lemma 9.6. $\Box$

Acknowledgement. Youshan Tao was supported by the National Natural Science
Foundation of China (No. 11861131003). The second author acknowledges support
of the Deutsche Forschungsgemeinschaft in the context of the project Emergence
of structures and advantages in cross-diffusion systems (No. 411007140, GZ: WI
3707/5-1).

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