Displaying similar documents to “Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena”

Numerical study on the blow-up rate to a quasilinear parabolic equation

Anada, Koichi, Ishiwata, Tetsuya, Ushijima, Takeo

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In this paper, we consider the blow-up solutions for a quasilinear parabolic partial differential equation u t = u 2 ( u x x + u ) . We numerically investigate the blow-up rates of these solutions by using a numerical method which is recently proposed by the authors [3].

Single-point blow-up for a semilinear parabolic system

Ph. Souplet (2009)

Journal of the European Mathematical Society

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We consider positive solutions of the system u t - Δ u = v p ; v t - Δ v = u q in a ball or in the whole space, with p , q > 1 . Relatively little is known on the blow-up set for semilinear parabolic systems and, up to now, no result was available for this basic system except for the very special case p = q . Here we prove single-point blow-up for a large class of radial decreasing solutions. This in particular solves a problem left open in a paper of A. Friedman and Y. Giga (1987). We also obtain lower pointwise estimates for...

Solvability problem for strong-nonlinear nondiagonal parabolic system

Arina A. Arkhipova (2002)

Mathematica Bohemica

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A class of q -nonlinear parabolic systems with a nondiagonal principal matrix and strong nonlinearities in the gradient is considered.We discuss the global in time solvability results of the classical initial boundary value problems in the case of two spatial variables. The systems with nonlinearities q ∈ ( 1 , 2 ) , q = 2 , q > 2 , are analyzed.

Liouville-type theorems and asymptotic behavior of nodal radial solutions of semilinear heat equations

Thomas Bartsch, Peter Poláčik, Pavol Quittner (2011)

Journal of the European Mathematical Society

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We prove a Liouville type theorem for sign-changing radial solutions of a subcritical semilinear heat equation u t = Δ u + u p - 1 u . We use this theorem to derive a priori bounds, decay estimates, and initial and final blow-up rates for radial solutions of rather general semilinear parabolic equations whose nonlinearities have a subcritical polynomial growth. Further consequences on the existence of steady states and time-periodic solutions are also shown.

Asymptotic analysis for a nonlinear parabolic equation on ℝ

Eva Fašangová (1998)

Commentationes Mathematicae Universitatis Carolinae

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We show that nonnegative solutions of u t - u x x + f ( u ) = 0 , x ∈ ℝ , t > 0 , u = α u ¯ , x ∈ ℝ , t = 0 , supp u ¯ compact either converge to zero, blow up in L 2 -norm, or converge to the ground state when t → ∞ , where the latter case is a threshold phenomenon when α > 0 varies. The proof is based on the fact that any bounded trajectory converges to a stationary solution. The function f is typically nonlinear but has a sublinear growth at infinity. We also show that for superlinear f it can happen that solutions converge to zero for any α > 0 , provided supp u ¯ is sufficiently small. ...

Blow-up of solutions for the non-Newtonian polytropic filtration equation with a generalized source

Jun Zhou (2016)

Annales Polonici Mathematici

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This paper deals with the blow-up properties of the non-Newtonian polytropic filtration equation u t - d i v ( | ∇ u m | p - 2 ∇ u m ) = f ( u ) with homogeneous Dirichlet boundary conditions. The blow-up conditions, upper and lower bounds of the blow-up time, and the blow-up rate are established by using the energy method and differential inequality techniques.