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Strong solvability for a class of nonlinear parabolic equations. (English) Zbl 0911.35115

The existence of strong solutions to the Cauchy-Dirichlet problem \[ a(x,t,u,u_x,u_{xx})-u_t=f(x,t,u,u_x)\text{ a.e. in }{\mathcal Q},\quad u=0\text{ on }\partial{\mathcal Q} \] in the rectangle \({\mathcal Q}=\{(x,t)\in(0,d)\times(0,T)\}\) with parabolic boundary \(\partial{\mathcal Q}=\{(x,0),x\in[0;d]\}\cup\{(0,t)\), \(t\in[0,T]\}\cup\{(d,t),t\in[0,T]\}\) is studied. The functions \(a(x,t,z,p,\xi)\) and \(f(x,t,z,p)\) are supposed to fulfill Carathéodory’s condition, \(a(x,t,z,p,\xi)\) is assumed to satisfy an ellipticity condition and \(f(x,t,z,p)\) is allowed to have quadratic growth with respect to the variable \(p\).
The existence of strong solutions to this problem is established through Aleksandrov-Bakel’man-Pucci type maximum principle and Lerey-Schauder’s fixed point theorem.

MSC:

35R05 PDEs with low regular coefficients and/or low regular data
35B45 A priori estimates in context of PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
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