Stability for evolution equations with variable growth
| Autor: | Shmarev, Sergey, Simsen, Jacson, Simsen, Mariza Stefanello |
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| Rok vydání: | 2021 |
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| Druh dokumentu: | Working Paper |
| Popis: | We study the character of dependence on the data and the nonlinear structure of the equation for the solutions of the homogeneous Dirichlet problem for the evolution $p(x,t)$-Laplacian with the nonlinear source \[ u_t-\Delta_{p(x,t)}u=f(x,t,u),\quad (x,t)\in Q=\Omega\times (0,T), \] where $\Omega$ is a bounded domain in $\mathbb{R}^n$, $n\geq 2$, and $p(x,t)$ is a given function $p(\cdot):Q\mapsto (\frac{2n}{n+2},p^+]$, $p^+<\infty$. It is shown that the solution is stable with respect to perturbations of the variable exponent $p(x,t)$, the nonlinear source term $f(x,t,u)$, and the initial data. We obtain quantitative estimates on the norm of the difference between two solutions in a variable Sobolev space through the norms of perturbations of the nonlinearity exponent and the data $u(x,0)$, $f$. Estimates on the rate of convergence of a sequence of solutions to the solution of the limit problem are derived. Comment: 18 pages |
| Databáze: | arXiv |
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