Displaying similar documents to “Solvability problem for strong-nonlinear nondiagonal parabolic system”

The Wolff gradient bound for degenerate parabolic equations

Tuomo Kuusi, Giuseppe Mingione (2014)

Journal of the European Mathematical Society

Similarity:

The spatial gradient of solutions to non-homogeneous and degenerate parabolic equations of p -Laplacean type can be pointwise estimated by natural Wolff potentials of the right hand side measure.

Continuability in time of smooth solutions of strong-nonlinear nondiagonal parabolic systems

Arina Arkhipova (2002)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Similarity:

A class of quasilinear parabolic systems with quadratic nonlinearities in the gradient is considered. It is assumed that the elliptic operator of a system has variational structure. In the multidimensional case, the behavior of solutions of the Cauchy-Dirichlet problem smooth on a time interval [ 0 , T ) is studied. Smooth extendibility of the solution up to t = T is proved, provided that “normilized local energies” of the solution are uniformly bounded on [ 0 , T ) . For the case where [ 0 , T ) determines the...

Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena

José M. Arrieta, Anibal Rodriguez-Bernal, Philippe Souplet (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Similarity:

We consider a one-dimensional semilinear parabolic equation with a gradient nonlinearity. We provide a complete classification of large time behavior of the classical solutions u : either the space derivative u x blows up in finite time (with u itself remaining bounded), or u is global and converges in C 1 norm to the unique steady state. The main difficulty is to prove C 1 boundedness of all global solutions. To do so, we explicitly compute a nontrivial Lyapunov functional by carrying out...

Cauchy-Neumann problem for a class of nondiagonal parabolic systems with quadratic growth nonlinearities I. On the continuability of smooth solutions

Arina A. Arkhipova (2000)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

A class of nonlinear parabolic systems with quadratic nonlinearities in the gradient (the case of two spatial variables) is considered. It is assumed that the elliptic operator of the system has a variational structure. The behavior of a smooth on a time interval [ 0 , T ) solution to the Cauchy-Neumann problem is studied. For the situation when the “local energies” of the solution are uniformly bounded on [ 0 , T ) , smooth extendibility of the solution up to t = T is proved. In the case when [ 0 , T ) defines...

Regularity problem for one class of nonlinear parabolic systems with non-smooth in time principal matrices

Arina A. Arkhipova, Jana Stará (2019)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Partial regularity of solutions to a class of second order nonlinear parabolic systems with non-smooth in time principal matrices is proved in the paper. The coefficients are assumed to be measurable and bounded in the time variable and VMO-smooth in the space variables uniformly with respect to time. To prove the result, we apply the so-called A ( t ) -caloric approximation method. The method was applied by the authors earlier to study regularity of quasilinear systems.