论文标题
具有可变增长的双相抛物线方程的强溶液
Strong solutions of the double phase parabolic equations with variable growth
论文作者
论文摘要
本文解决了与可变增长的运营商的双相方程的同质迪里奇问题有关的存在的问题和强大解决方案的独特性： \ [u_t-div \ left（| \ nabla u |^{p（z）-2} \ nabla u+ a（z）| \ nabla u |^{q（z）-2} \ nabla u \ right） \ Mathbb {r}^n $，$ n \ geq 2 $，是一个有界域，具有边界$ \partialΩ f（z，v）= f_0（z）+b（z）| v |^{σ（z）-2} v。 \ \]可变指数$ p $，$ q $，$σ$在$ \ bar {q} _t $，$ p $，$ q $上定义的功能是Lipschitz-continuule和\ [\ dfrac {\ dfrac {2n} {n +2} {n +2} {n +2} <p^ - \ leq P（z） {\ frac {r} {2}} \ \ \ text {带有$ 0 <r <r <r <r^\ ast = \ frac {4p^ - } {2n + p^ - （n + 2）} $，\ quad $ p^ - = = \ min _} \]我们在功能上找到条件$ f_0 $，$ a $，$ b $，$σ$和$ u_0 $足以使存在具有以下全局规则性和集成性属性的独特强度解决方案：\ [\ [\ [split} split} u_t \ in L^{2}（q_t}（q_t）（q_t），\ quad），\ quad＆q^$ | l^{\ infty}（0，t; t; l^1（ω））$，$ s（z）= \ max \ {2，p（z）\} $} $}，＆| \ nabla u |^{p（z）+δ}+Δ}+Δ}+u^1（q_t）\ quad \ quad \ quad \ fort $ 0 $ 0 < {split} \]与正则通量的方程式建立了相同的结果 \ [（ε^2+| \ nabla u |^2）^{\ frac {p（z）-2} {2}} {2}} \ nabla u+a（z）（ε^2+| \ nabla u |^2） \]
This paper addresses the questions of existence and uniqueness of strong solutions to the homogeneous Dirichlet problem for the double phase equation with operators of variable growth: \[ u_t - div \left(|\nabla u|^{p(z)-2} \nabla u+ a(z) |\nabla u|^{q(z)-2} \nabla u \right) = F(z,u) \quad \text{in $Q_T=Ω\times (0,T)$} \] where $Ω\subset \mathbb{R}^N$, $N \geq 2$, is a bounded domain with the boundary $\partialΩ\in C^2$, $z=(x,t)\in Q_T$, $a:\bar Q_T \mapsto \mathbb{R}$ is a given nonnegative coefficient, and the nonlinear source term has the form \[ F(z,v)=f_0(z)+b(z)|v|^{σ(z)-2}v. \] The variable exponents $p$, $q$, $σ$ are given functions defined on $\bar{Q}_T$, $p$, $q$ are Lipschitz-continuous and \[ \dfrac{2N}{N+2}<p^-\leq p(z) \leq q(z) < p(z) + {\frac{r}{2}} \ \ \text{with $0<r<r^\ast=\frac{4p^-}{2N + p^-(N+2)}$,\quad $p^-=\min_{\bar{Q}_T}p(z)$}. \] We find conditions on the functions $f_0$, $a$, $b$, $σ$ and $ u_0$ sufficient for the existence of a unique strong solution with the following global regularity and integrability properties: \[ \begin{split} u_t \in L^{2}(Q_T),\quad & \text{$|\nabla u|^{s(z)} \in L^{\infty}(0,T;L^1(Ω))$ with $ s(z)=\max\{2,p(z)\}$}, & |\nabla u|^{p(z)+δ}\in L^1(Q_T)\quad \text{for every $0<δ< r^*$}. {split} \] The same results are established for the equation with the regularized flux \[ (ε^2+|\nabla u|^2)^{\frac{p(z)-2}{2}}\nabla u + a(z) (ε^2+|\nabla u|^2)^{\frac{q(z)-2}{2}}\nabla u, \qquad ε>0. \]
