Lp-estimates of solutions of some nonlinear degenerate diffusion equations
Mitsuhiro Nakao · Journal of the Mathematical Society of Japan · 1985
The object of this paper is to show the existence, uniqueness and $L^{p}$ estimates (including $p=\infty$ ) of global solutions of some nonlinear degenerate diffusion equations.The first problem we are concerned with is the following initial-boundary value problem of the perturbed porous medium equation;$\frac{\partial}{\partial t}u-\Delta u^{m+1}+f(x, t, u)=0$ in $\Omega\cross(0, \infty)$ $(P_{1})$ $u(x, 0)=u_{0}(\geqq 0)$ , $u|_{\partial\Omega}=0$ and $u\geqq 0$ where $\Omega$ is a bounded domain in $R^{N}$ with smooth boundary $\partial\Omega(C^{3}$ class is sufficient), $m$ is a positive constant and $f(x, t, u)$ is a function satisfying; ASSUMPTION 1. (i) $f(x, t, u)$ is locally H\"older continuous in $\overline{\Omega}\cross R^{+}\cross R^{+}$ $(R^{+}=[0, \infty))$ and locally Lipschitz continuous with respect to $u$ uniformly in $(x, t)$ .(ii) $f(x, t, u)\geqq-C_{0}u^{1+\alpha}$ on $\overline{\Omega}\cross R^{+}\cross R^{+}$ for some $C_{0}>0$ and $\alpha\geqq 0$ .It should be noted that the theory of nonlinear semi-groups does not apply to $(P_{1})$ for the existence of global solution since we do not assume that $f(x, t, u)$ is monotone with respect to $u$ .To treat the problem $(P_{1})$ it is convenient to compare it with the problem; $\frac{\partial}{\partial t}u-\Delta u^{m+1}-C_{0}u^{1+\alpha}=0$ in $\Omega\cross(0, \infty)$ $(P_{2})$ $u(x, 0)=u_{0}(\geqq 0)$ , $u|_{\partial\Omega}=0$ and $u\geqq 0$ .Recently in [13] we have discussed the existence, nonexistence and some