Nonlinear diffusion equations driven by the \(p(\cdot)\)-Laplacian. (English) Zbl 1264.35136
The authors considered the following initial-boundary value problem:
\[
\partial_tu=\nabla\cdot(|\nabla\phi(x)|^{p(x)-2}\nabla\phi(x))+f,\qquad\text{in }\Omega\times (0,\infty),
\]
\[ u=0,\qquad\text{on }\partial\Omega\times (0,\infty), \]
\[ u(\cdot,0)=u_0,\qquad\text{in }\Omega, \] where \(\Omega\subset{\mathbb R}^N\) is a bounded domain with smooth boundary \(\partial\Omega\), \(f\) and \(u_0\) are given functions.
The authors investigated the asymptotic behaviors of the solutions of the problem as \(t\rightarrow 0\) when \(f\equiv0\), and also found out the limiting behavior of the solutions as \(p_n(\cdot)\rightarrow\infty\).
\[ u=0,\qquad\text{on }\partial\Omega\times (0,\infty), \]
\[ u(\cdot,0)=u_0,\qquad\text{in }\Omega, \] where \(\Omega\subset{\mathbb R}^N\) is a bounded domain with smooth boundary \(\partial\Omega\), \(f\) and \(u_0\) are given functions.
The authors investigated the asymptotic behaviors of the solutions of the problem as \(t\rightarrow 0\) when \(f\equiv0\), and also found out the limiting behavior of the solutions as \(p_n(\cdot)\rightarrow\infty\).
Reviewer: Yuanyuan Ke (Beijing)
MSC:
| 35K92 | Quasilinear parabolic equations with \(p\)-Laplacian |
| 35K55 | Nonlinear parabolic equations |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
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