Estimates of solutions of quasilinear systems of parabolic type. (Russian) Zbl 0706.35064

The author obtains bounds - that do not assume the construction of a Lyapunov functional - for strong solutions of the system \[ du/dt=\Phi(S)u+F(u,t),\;t\geq 0,\;u\in L^ 2(\Lambda,\mu,{\mathbb{C}}^ n), \] \(\Lambda\) a set with a finite measure \(\mu\), \(u(t)\in {\mathbb{C}}^ n\), \(\Phi(S)=\sum^{p}_{k=0}a_ kS^ k\), \(a_ k\), \(k=1,...,p\), \((n\times n)\) matrices, S a normal operator on \(L^ 2\) that commutes with the operators on \(L^ 2\) given by these \(a_ k\) matrices. With D \(=\) domain of \(\Phi(S), F: D\times [0,\infty)\mapsto L^ 2\) and satisfies \[ \| S^{-\eta} F(u,t)\|_{L^ 2}\leq q(t)\| u\|_{L^ 2}+\nu (t),\quad t\geq 0,\quad \| u\|_{L^ 2}\leq r, \] \(0\leq \eta <p\), q and \(\nu\) measurable and positive.
Reviewer: J.E.Bouillet


35K55 Nonlinear parabolic equations
35B45 A priori estimates in context of PDEs
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations