Quasilinear parabolic equations containing a Volterra operator in the coefficients

G. I. Laptev

Sbornik. Mathematics, Tome 64 (1989) no. 2, pp. 527-542

Résumé

Conditions are established for solvability in the large of the first initial-boundary value problem in a bounded domain $\Omega\subset R^n$ for the equation $$ u_t+(-1^m)\sum_{|\alpha|=m}D^\alpha\biggl[a_\alpha\biggl(\int_0^t|D^\alpha u|^q\,dt\biggr)|D^\alpha u|^{q-2}D^\alpha u\biggr]=f, $$ where $q\geqslant2$. It contains the integral of the unknown function in the coefficients. The problem is regarded as an evolution equation of the form $u'+Au=f$. Conditions of polynomial growth are imposed on the functions $a_\alpha(s)$: $$ a_0s^r\leqslant a_\alpha(s)\leqslant a_1s^r+a_2\qquad(a_i>0;\ r>0). $$ The space $\mathring W_p^m(\Omega;L^q(0,T))$, is constructed, where $p=q(1+r)$; the operator $A$ is coercive in this space. Under the additional assumption that the functions $a_\alpha(s)$ are convex (which corresponds to exponents $0$) it is proved that $A$ is a monotone operator and the corresponding evolution equation is solvable. Bibliography: 6 titles.