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

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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

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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

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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

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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...

Boundedness of solutions to parabolic-elliptic chemotaxis-growth systems with signal-dependent sensitivity

Kentarou Fujie, Tomomi Yokota (2014)

Mathematica Bohemica

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This paper deals with parabolic-elliptic chemotaxis systems with the sensitivity function χ ( v ) and the growth term f ( u ) under homogeneous Neumann boundary conditions in a smooth bounded domain. Here it is assumed that 0 < χ ( v ) ≤ χ 0 / v k ( k ≥ 1 , χ 0 > 0 ) and λ 1 - μ 1 u ≤ f ( u ) ≤ λ 2 - μ 2 u ( λ 1 , λ 2 , μ 1 , μ 2 > 0 ) . It is shown that if χ 0 is sufficiently small, then the system has a unique global-in-time classical solution that is uniformly bounded. This boundedness result is a generalization of a recent result by K. Fujie, M. Winkler, T. Yokota. ...