On the asymptotic behaviour of solutions of nonlinear parabolic equations (Q793224)

We study the asymptotic behaviour of classical solutions of nonlinear parabolic equations \[ -u_ t+a_{ij}(x,t,u,\nabla u)u_{x_ ix_ j}+f(x,t,u,\nabla u)=0\quad on\quad \Omega \times {\mathbb{R}}^+ \] with, say, \(u|_{\partial \Omega}=0\) and \(u(x,0)=u_ 0(x)\). Assuming w.l.o.g. that u converges to the trivial steady state we are interested in the exact decay rate for \(t\to \infty\). Under some natural conditions (e.g. quadratic growth of f(x,t,u,p) with respect to p for estimating the gradient) it is shown, that \[ | u(x,t)| +| u_ t(x,t)| +| \nabla u(x,t)| \leq Ce^{-\lambda_ 0t}\quad on\quad {\bar \Omega}\times {\mathbb{R}}^+ \] where \(\lambda_ 0>0\) is the first eigenvalue of the linearized equation \(-Lw=-A_{ij}w_{x_ ix_ j}- A_ iw_{x_ i}-A_ 0w,\) assuming \(|(\partial f/\partial u)(x,t,0,0)-A_ 0(x)| \in L_ 1({\mathbb{R}}^+), A_ 0\leq 0\), and similar conditions for the other terms. These estimates are sharp in general. The results are applied to the solutions of two reaction- diffusion systems.