Geodesic
Quasilinear parabolic equations containing a Volterra operator in the coefficients
Sbornik. Mathematics, Tome 64 (1989) no. 2, pp. 527-542 
G. I. Laptev. Quasilinear parabolic equations containing a Volterra operator in the coefficients. Sbornik. Mathematics, Tome 64 (1989) no. 2, pp. 527-542. http://geodesic.mathdoc.fr/item/SM_1989_64_2_a15/
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     author = {G. I. Laptev},
     title = {Quasilinear parabolic equations containing a {Volterra} operator in the coefficients},
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     year = {1989},
     volume = {64},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1989_64_2_a15/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

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