Nonlinear diffusion equations driven by the \(p(\cdot)\)-Laplacian
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 \(p_n(\cdot)\rightarrow\infty\).
scientific article; zbMATH DE number 6138201
scientific article