Remarks on global solutions to the initial-boundary value problem for quasilinear degenerate parabolic equations with a nonlinear source term

Abstract

We give an existence theorem of global solution to the initial-boundary value problem for $u_{t}-\operatorname{div}{\sigma(|\nabla u|^2)\nabla u}=f(u)$ under some smallness conditions on the initial data, where $\sigma (v^2)$ is a positive function of $v^{2}\neq 0$ admitting the degeneracy property $\sigma(0)=0$. We are interested in the case where $\sigma(v^{2})$ has no exponent $m \geq 0$ such that $\sigma(v^2) \geq k_0|v|^m , k_0 \gt 0$. A typical example is $\sigma(v^2)=\operatorname{log}(1+v^2)$. $f(u)$ is a function like $f=|u|^{\alpha} u$. A decay estimate for $|\nabla u(t)|_{\infty}$ is also given.