大时间存在性在具有轻微温度依赖参数的热粘弹性演化问题中
Large time existence in a thermoviscoelastic evolution problem with mildly temperature-dependent parameters
AI总结：
本文研究了具有轻微温度依赖参数的热粘弹性演化问题，证明了经典解的存在时间可以任意大。
摘要：
我们考虑以下方程组：u_{tt} = (γ(Θ) u_{xt})_x + a (γ(Θ) u_x)_x +(f(Θ))_x，Θ_t = DΘ_{xx} + Γ(Θ) u_{xt}^2 + F(Θ) u_{xt}，在Ω⊂R的有界区间内，u满足Neumann边界条件，Θ满足Dirichlet边界条件。该模型是经典系统在Kelvin-Voigt材料定律下描述应变和温度演化问题的推广，其中γ≡Γ且f≡F。该模型的不同变体已经有过分析，本文基于已知结果，即对于合适光滑的初始数据、任意a>0、D>0以及γ,f∈C^2([0,∞))以及Γ,F∈C^1([0,∞))，且γ>0、Γ≥0和F(0)=0的情况下，经典解的存在性是局部时间的。我们的工作聚焦于证明经典解的存在时间可以任意大，假设γ和f具有亚线性温度依赖性，并进一步|F(s)|≤C_F(1+s)^α，其中C_F>0且α∈(0,1)。特别地，对于任何给定的T_*,初始质量M和0<γ<γ，存在一个常数δ_*(M,T_*,a,D,Ω,γ,γ,C_F,α)>0，使得如果γ≤γ且0≤Γ≤γ以及γ'在L^∞([0,∞))中的范数≤δ_*且f'在L^∞([0,∞))中的范数≤δ_*成立，则经典解到(★)的最大存在时间超过T_*。
英文摘要：
We consider \begin{align*} \label{HS} \left\{ \begin{array}{l} u_{tt} = (γ(Θ) u_{xt})_x + a (γ(Θ) u_x)_x +(f(Θ))_x, \\[1mm] Θ_t = DΘ_{xx} + Γ(Θ) u_{xt}^2 + F(Θ) u_{xt}, \end{array}\right. \qquad \qquad (\star) \end{align*} under Neumann boundary conditions for $u$ and Dirichlet boundary conditions for $Θ$ in a bounded interval $Ω\subset\mathbb{R}$. \abs This model is a generalization of the classical system for the description of strain and temperature evolution in a thermo-viscoelastic material following a Kelvin-Voigt material law, in which $γ\equiv Γ$ and $f\equiv F$. Different variations of this model have already been analyzed in the past and the present study draws upon a known result concerning the existence of classical solutions, which are local in time, for suitably smooth initial data, arbitrary $a>0$, $D>0$ and $γ,f\in C^2([0,\infty))$ as well as $Γ,F\in C^1([0,\infty))$ with $γ>0,Γ\ge0$ and $F(0)=0$. Our work focuses on proving that existence times for classical solutions can be arbitrarily large, assuming sublinear temperature dependencies of $γ$ and $f$, and further $|F(s)|\le C_F(1+s)^α$ for some $C_F>0$ and $α\in(0,1)$. In particular, for any given $T_\star$, initial mass $M$ and $0<\underlineγ<\overlineγ$, there exists a constant $δ_\star(M,T_\star,a,D, Ω, \underlineγ, \overlineγ,C_F,α)>0$, such that if $$\underlineγ\leγ\le \overlineγ\quad\mbox{ and }\quad 0\le Γ\le \overlineγ\quad \mbox{ as well as } \quad\|γ'\|_{L^\infty([0,\infty))}\le δ_\star \quad \mbox{ and }\quad \|f'\|_{L^\infty([0,\infty))}\le δ_\star $$ hold, the maximal existence time of the classical solution to $(\star)$ surpasses $T_\star$.
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