passing to the limit in heat equation (2.66). Let us suppose that $\psi_{2}\in
C_{0}^{\infty}([0,{\mathfrak{T}}])$, then (2.66) yields

$\begin{split}&\int_{0}^{\mathfrak{T}}\int_{\Omega}\theta^{k}_{t}\,\psi_{2}(t)v\,{\mathrm{d}}x\,{\mathrm{d}}t+\int_{0}^{\mathfrak{T}}\int_{\Omega}\nabla\theta^{k}\,\psi_{2}(t)\nabla
v\,{\mathrm{d}}x\,{\mathrm{d}}t\\\\[4.30554pt] &\hskip
8.61108pt+\int_{0}^{\mathfrak{T}}\int_{\Omega}f\big{(}{\mathcal{T}}_{k}(\tilde{\theta}+\theta^{k})\big{)}\mathrm{div}(\tilde{u}_{t}+u^{k}_{t})\,\psi_{2}(t)v\,{\mathrm{d}}x\,{\mathrm{d}}t\\\\[4.30554pt]
=&\int_{0}^{\mathfrak{T}}\int_{\Omega}{\mathcal{T}}_{k}\big{(}\big{\\{}|\operatorname{dev}(T^{k})|-\beta(\theta^{k}+\tilde{\theta})\big{\\}}^{r}_{+}|\operatorname{dev}(T^{k})|\big{)}\,\psi_{2}(t)v\,{\mathrm{d}}x\,{\mathrm{d}}t\,.\end{split}$
(3.41)

Again (3.25), Remark 3.4 and Lemma 3.12 give us

$\begin{split}&\int_{0}^{\mathfrak{T}}\int_{\Omega}(\theta+\tilde{\theta})_{t}\,\phi\,{\mathrm{d}}x\,{\mathrm{d}}t+\int_{0}^{\mathfrak{T}}\int_{\Omega}\nabla(\theta+\tilde{\theta})\,\phi\,{\mathrm{d}}x\,{\mathrm{d}}t\\\\[4.30554pt]
&\hskip
8.61108pt+\int_{0}^{\mathfrak{T}}\int_{\Omega}f\big{(}\tilde{\theta}+\theta)\mathrm{div}(\tilde{u}_{t}+u_{t})\,\phi\,{\mathrm{d}}x\,{\mathrm{d}}t\\\\[4.30554pt]
&=\int_{0}^{\mathfrak{T}}\int_{\Omega}\big{\\{}|\operatorname{dev}(T)|-\beta(\theta+\tilde{\theta})\big{\\}}^{r}_{+}|\operatorname{dev}(T)|\,\phi\,{\mathrm{d}}x\,{\mathrm{d}}t+\int_{0}^{\mathfrak{T}}\int_{\Omega}g_{\theta}\,\phi\,{\mathrm{d}}S(x)\,{\mathrm{d}}t\end{split}$
(3.42)

for all $\phi\in C^{\infty}([0,{\mathfrak{T}}];C^{\infty}(\overline{\Omega}))$
and the function $\tilde{\theta}$ is the solution of (2.2). To complete the
proof of the main result, we need to make sense of the initial condition for
the temperature function. The following lemma ensures that the initial
condition for temperature is satisfied in the standard sense.

######  Lemma 3.13.

The sequence $\\{\theta^{k}\\}_{k>0}$ converges strongly to $\theta$ in
$C([0,{\mathfrak{T}}];L^{1}(\Omega))$.

###### Proof.

The idea of the proof is taken from Lemma 1 of [11]. Fix $M>0$, then selecting
the test function $v={\mathcal{T}}_{M}(\theta^{k}-\theta^{l})$ in the
difference of the heat equations for $\theta^{k}$ and $\theta^{l}$ (see (2.66)
if necessary) we get

$\begin{split}\int_{\Omega}&\varphi_{M}(\theta^{k}-\theta^{l})(t)\,{\mathrm{d}}x+\int_{0}^{t}\int_{\Omega}|\nabla{\mathcal{T}}_{M}(\theta^{k}-\theta^{l})|^{2}\,{\mathrm{d}}x\,{\mathrm{d}}\tau\\\\[4.30554pt]
&=\int_{0}^{t}\int_{\Omega}\Big{(}G^{k}-G^{l}\Big{)}{\mathcal{T}}_{M}(\theta^{k}-\theta^{l})\,{\mathrm{d}}x\,{\mathrm{d}}\tau+\int_{\Omega}\varphi_{M}(\theta^{k}-\theta^{l})(0)\,{\mathrm{d}}x\,,\end{split}$
(3.43)

for almost every $t\leq{\mathfrak{T}}$, where

$G^{i}=f\big{(}{\mathcal{T}}_{i}(\theta^{i}+\tilde{\theta})\big{)}\mathrm{div}\,u^{i}_{t}+{\mathcal{T}}_{k}\Big{(}\big{\\{}|\operatorname{dev}(T^{i})|-\beta(\theta^{i}+\tilde{\theta})\big{\\}}^{r}_{+}|\operatorname{dev}(T^{i})|\Big{)}$
(3.44)

for $i=k,\,l$. Observe that the function
${\mathcal{T}}_{M}(\theta^{k}-\theta^{l})\rightarrow 0$ for almost all
$(x,\tau)\in\Omega\times(0,t)$ (as $k,\,l\rightarrow\infty$). Additionally,
${\mathcal{T}}_{M}(\theta^{k}-\theta^{l})$ is uniformly bounded. The Egorov
Theorem entails that for every $\nu>0$, there exists a measurable subset $A$
of $\Omega\times(0,t)$ such that $|A|<\nu$ and
${\mathcal{T}}_{M}(\theta^{k}-\theta^{l})$ converges to $0$ uniformly on
$\Omega\times(0,t)\setminus A$. Therefore,

$\begin{split}\int_{\Omega}&\varphi_{M}(\theta^{k}-\theta^{l})(t)\,{\mathrm{d}}x\leq\int_{0}^{t}\int_{\Omega}\Big{(}G^{k}-G^{l}\Big{)}{\mathcal{T}}_{M}(\theta^{k}-\theta^{l})\,{\mathrm{d}}x\,{\mathrm{d}}\tau+\int_{\Omega}\varphi_{M}(\theta^{k}-\theta^{l})(0)\,{\mathrm{d}}x\\\\[4.30554pt]
&=\int_{\Omega\times(0,t)\setminus
B}\Big{(}G^{k}-G^{l}\Big{)}{\mathcal{T}}_{M}(\theta^{k}-\theta^{l})\,{\mathrm{d}}x\,{\mathrm{d}}\tau\\\\[4.30554pt]
&\hskip
8.61108pt+\int_{B}\Big{(}G^{k}-G^{l}\Big{)}{\mathcal{T}}_{M}(\theta^{k}-\theta^{l}){\mathrm{d}}x\,{\mathrm{d}}\tau+\int_{\Omega}\varphi_{M}(\theta^{k}-\theta^{l})(0)\,{\mathrm{d}}x\,\\\\[4.30554pt]
\end{split}$ (3.45)

From Lemma 4.5, we know that the sequence $G^{k}$ converges weakly in
$L^{1}(0,{\mathfrak{T}};L^{1}(\Omega))$ (it is bounded), hence the first
integral on the right hand-side of (3.45) tends to zero. Notice that
$(\theta^{k}-\theta^{l})(0)={\mathcal{T}}_{k}(\theta_{0})-{\mathcal{T}}_{l}(\theta_{0})\rightarrow
0$ in $L^{1}(\Omega)$ as $k,\,l\rightarrow\infty$. The functions $G^{k}$ and
$G^{l}$ are uniformly integrable, thus the second integral on the right hand-
side of (3.45) is arbitrarily small. Moreover, for $r\in{\mathbb{R}}$

$\varphi_{M}(r)\geq\left\\{\begin{array}[]{ll}\frac{1}{2}|r|^{2}&\textrm{if}\quad|r|\leq
M\,,\\\\[4.30554pt] \frac{1}{2}M|r|&\textrm{if}\quad|r|>M\\\
\end{array}\right.$

and the sequence $\\{\theta^{k}\\}_{k>0}$ is a Cauchy sequence in the space
$C([0,{\mathfrak{T}}];L^{1}(\Omega))$. ∎

######  Remark 3.14.

From Lemma 3.13, we know that the sequence $\theta^{k}(\cdot,0)$ convergences
to $\theta(\cdot,0)$ in $L^{1}(\Omega)$. It is not hard to observe that
$\theta^{k}(\cdot,0)={\mathcal{T}}_{k}(\theta_{0})\rightarrow\theta_{0}$ in
$L^{1}(\Omega)$. Hence $\theta_{0}=\theta(\cdot,0)$ and the initial condition
for the temperature function is satisfied in a classical sense.

The Remark 3.14 ends with the proof of the existence of solutions for the
considered system (1.11) with the boundary conditions (1.12) and the initial
conditions (1.13).

## References

  * [1] H.-D. Alber.  Materials with memory, volume 1682 of Lecture Notes in Mathematics.  Springer-Verlag, Berlin, 1998.  Initial-boundary value problems for constitutive equations with internal variables.
  * [2] H.-D. Alber and K. Chełmiński.  Quasistatic problems in viscoplasticity theory. I. Models with linear hardening.  In Operator theoretical methods and applications to mathematical physics, volume 147 of Oper. Theory Adv. Appl., pages 105–129. Birkhäuser, Basel, 2004.
  * [3] J. Alibert and G. Bouchitté.  Non-uniform integrability and generalized young measures.  Journal of Convex Analysis, 4(1):129–147, 1997.
  * [4] H. W. Alt.  Lineare Funktionalanalysis.  Springer-Verlag, 2006.
  * [5] J.-P. Aubin and H. Frankowska.  Set-valued analysis.  Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, 2009\.  Reprint of the 1990 edition [MR1048347].
  * [6] L. Bartczak.  Mathematical analysis of a thermo-visco-plastic model with Bodner-Partom constitutive equations.  J. Math. Anal. Appl., 385(2):961–974, 2012.
  * [7] L. Bartczak.  Analysis of a thermo-viscoplastic model with Lipschitz continuous constitutive equations.  Math. Methods Appl. Sci., 37(17):2597–2614, 2014.
  * [8] S. Bartels and T. Roubíček.  Thermo-visco-elasticity with rate-independent plasticity in isotropic materials undergoing thermal expansion.  ESAIM Math. Model. Numer. Anal., 45(3):477–504, 2011.
  * [9] A. Bensoussan and J. Frehse.  Asymptotic behaviour of the time dependent Norton-Hoff law in plasticity theory and $H^{1}$ regularity.  Comment. Math. Univ. Carolin., 37(2):285–304, 1996.
  * [10] A. Bensoussan and J. Frehse.  Regularity Results for Nonlinear Elliptic Systems and Applications.  Applied Mathematical Sciences, Vol. 151, Springer Verlag Berlin, 2002\.
  * [11] D. Blanchard.  Truncations and monotonicity methods for parabolic equations.  Nonlinear Anal., 21(10):725–743, 1993.
  * [12] D. Blanchard and O. Guibé.  Existence d’une solution pour un système non linéaire en thermoviscoélasticité.  C. R. Acad. Sci. Paris Sér. I Math., 325(10):1125–1130, 1997\.
  * [13] D. Blanchard and O. Guibé.  Existence of a solution for a nonlinear system in thermoviscoelasticity.  Adv. Differential Equations, 5(10-12):1221–1252, 2000.
  * [14] D. Blanchard and F. Murat.  Renormalised solutions of nonlinear parabolic problems with L1 data: existence and uniqueness.  Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 127:1137–1152, 1997.
  * [15] L. Boccardo and T. Gallouët.  Non-linear elliptic and parabolic equations involving measure data.  Journal of Functional Analysis, 87(1):149 – 169, 1989.
  * [16] H. Brezis.  Functional Analysis, Sobolev Spaces and Partial Differential Equations.  Universitext. Springer-Verlag New York, 2011.
  * [17] K. Chełmiński and P. Gwiazda.  Nonhomogeneous initial-boundary value problems for coercive and self-controlling models of monotone type.  Contin. Mech. Thermodyn., 12(4):217–234, 2000.
  * [18] K. Chełmiński and P. Gwiazda.  On the model of Bodner-Partom with nonhomogeneous boundary data.  Math. Nachr., 214:5–23, 2000.
  * [19] K. Chełmiński and P. Gwiazda.  Convergence of coercive approximations for strictly monotone quasistatic models in inelastic deformation theory.  Mathematical Methods in the Applied Sciences, 30(12):1357–1374, 2007.
  * [20] K. Chełmiński, P. Neff, and S. Owczarek.  The Armstrong-Frederick cyclic hardening plasticity model with Cosserat effects.  J. Diff. Equations, 256(11):3497–3523, 2014.
  * [21] K. Chełmiński, P. Neff, and S. Owczarek.  A first regularity result for the Armstrong-Frederick cyclic hardening plasticity model with Cosserat effects.  J. Math. Anal. Appl., 423(1):283 – 304, 2015.
  * [22] K. Chełmiński and S. Owczarek.  Renormalised solutions in thermo-visco-plasticity for a Norton-Hoff type model. Part I: The truncated case.  Nonlinear Analysis: Real World Applications, 28:140–152, 2016.
  * [23] K. Chełmiński and S. Owczarek.  Renormalised solutions in thermo-visco-plasticity for a Norton-Hoff type model. Part II: the limit case.  Nonlinear Anal. Real World Appl., 31:643–660, 2016.
  * [24] K. Chełmiński and R. Racke.  Mathematical analysis of thermoplasticity with linear kinematic hardening.  J. Appl. Anal., 12(1):37–57, 2006.
  * [25] A. Dall’Aglio and L. Orsina.  Nonlinear parabolic equations with natural growth conditions and $L^{1}$ data.  Nonlinear Anal., 27(1):59–73, 1996.
  * [26] G. Duvaut and J.-L. Lions.  Les inéquations en mécanique et en physique.  Dunod, Paris, 1972.  Travaux et Recherches Mathématiques, No. 21.
  * [27] P. Gwiazda, F. Klawe, and S. Owczarek.  Thermo-visco-elasticity for Norton-Hoff-type models with homogeneous thermal expansion.  Nonlinear Anal. Real World Appl., 40:337–360, 2018.
  * [28] P. Gwiazda, F. Z. Klawe, and A. Świerczewska-Gwiazda.  Thermo-visco-elasticity for the Mróz model in the framework of thermodynamically complete systems.  Discrete Contin. Dyn. Syst. Ser. S, 7(5):981–991, 2014.
  * [29] P. Gwiazda, F. Z. Klawe, and A. Świerczewska-Gwiazda.  Thermo-visco-elasticity for Norton-Hoff-type models.  Nonlinear Analysis: Real World Applications, 26:199 – 228, 2015\.
  * [30] P. Gwiazda and A. Świerczewska.  Large eddy simulation turbulence model with young measures.  Applied Mathematics Letters, 18(8):923–929, 2005.
  * [31] P. Haupt.  Continuum mechanics and theory of materials.  Advanced Texts in Physics. Springer-Verlag, Berlin, 2000.  Translated from the German by Joan A. Kurth.
  * [32] R. Herzog, C. Meyer, and A. Stötzner.  Existence of solutions of a thermoviscoplastic model and associated optimal control problems.  Nonlinear Analysis: Real World Applications, 35:75–101, 2017.
  * [33] I. Ionescu and M. Sofonea.  Functional and Numerical Methods in Viscoplasticity.  Oxford mathematical monographs. Oxford University Press, 1993.
  * [34] F. Z. Klawe.  Thermo-visco-elasticity for models with growth conditions in Orlicz spaces.  Topol. Methods Nonlinear Anal., 43(2):297 – 322, 2016.
  * [35] F. Z. Klawe and S. Owczarek.  Thermo-visco-elasticity for models with non-homogeneous thermal expansion and with growth conditions in orlicz spaces.  Submitted to: Nonlinear Analysis: Theory, Methods and Applications, 2020.
  * [36] J. Málek, J. Nečas, M. Rokyta, and M. Růžicka.  Weak and measure-valued solutions to evolutionary PDEs.  Chapman & Hall, London, 1996.
  * [37] R. v. Mises.  Mechanik der festen Körper im plastisch- deformablen Zustand.  Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 1913:582–592, 1913.
  * [38] S. Müller.  Variational models for microstructure and phase transitions, pages 85–210.  Springer Berlin Heidelberg, Berlin, Heidelberg, 1999.
  * [39] S. Owczarek.  Convergence of a monotonisation procedure for a non-monotone quasi-static model in poroplasticity.  J. Math. Anal. Appl., 364(2):1541–1563, 2010.
  * [40] S. Owczarek and L. Bartczak.  On renormalized solutions for thermomechanical problems in perfect plasticity with damping forces.  Math. Mech. Solids, 24(4):1030–1053, 2019.
  * [41] S. Owczarek and F. Z. Klawe.  Thermo-visco-elasticity for Norton-Hoff-type models with Cosserat effects.  Math. Meth. Appl. Sciences, 39:4629–4641, 2016.
  * [42] T. Roubíček.  Nonlinear partial differential equations with applications, volume 153 of International Series of Numerical Mathematics.  Birkhäuser Verlag, Basel, 2005.
  * [43] T. Roubíček.  Thermodynamics of perfect plasticity.  Discrete Contin. Dyn. Syst. Ser. S, 6(1):193–214, 2013.
  * [44] T. Roubíček.  Thermo-visco-elasticity at small strains with $L^{1}$-data.  Quart. Appl. Math., 67(1):47–71, 2009.
  * [45] W. Strauss.  Partial differential equations.  Wiley, 2007.
  * [46] R. Temam.  Problèmes mathématiques en plasticité, volume 12 of Méthodes Mathématiques de l’Informatique [Mathematical Methods of Information Science].  Gauthier-Villars, Montrouge, 1983.
  * [47] R. Temam.  A generalized Norton-Hoff model and the prandtl-reuss law of plasticity.  Archive for Rational Mechanics and Analysis, 95(2):137–183, 1986\.
  * [48] T. Valent.  Boundary Value Problems of Finite Elasticity: Local Theorems on Existence, Uniqueness, and Analytic Dependence on Data.  Springer Publishing Company, Incorporated, 1st edition, 2011.
  * [49] E. Zeidler.  Nonlinear functional analysis and its applications. II/B.  Springer-Verlag, New York, 1990.  Nonlinear monotone operators, Translated from the German by the author and Leo F. Boron.