Self-Similar Subsolutions and Blowup for Nonlinear Parabolic Equations

Abstract

For a wide class of nonlinear parabolic equations of the form u y ⌬ u s t Ž . F u, ٌu , we prove the nonexistence of global solutions for large initial data. We also estimate the maximal existence time. To do so we use a method of comparison with suitable blowing up self-similar subsolutions. As a consequence, we improve several known results on u y ⌬ u s u p , on generalized Burgers' equations, and on t other semilinear equations. This method can also apply to degenerate equations of porous medium type and provides a unified treatment for a large class of problems, both semilinear and quasilinear.

Blow-Up of Solutions for a Class of Reaction-Diffusion Equations with a Gradient Term under Nonlinear Boundary Condition

“…The blow-up phenomenon of (1) with linear boundary conditions have been extensively studied over the past decades. In [14,15], Souplet discussed the following equations with Dirichlet boundary conditions…”

L∞ estimates of solutions for the quasilinear parabolic equation with nonlinear gradient term and L1 data

“…Proof of Theorem 2.4 Noticing that the estimate constant C 0 in (3.12)-(3.13) and (3.27)-(3.28) is independent of i, we have from the standard compact argument as in [1,13,14] that there exists a subsequence (still denoted by u i ) and a function…”

Relative stabilization of solutions of a degenerating parabolic equation with nonlinear lower-order terms

“…180-185], it is meaningful to consider nonnegative functions u and f ), the input v ∈ R 1 is a control, and the parameters a, c, α, and λ are real numbers such that ac = 0, α > 1, and λ > 0. A function v(x, t), (x, t) ∈S, belongs to the set V of admissible controls if v is continuous inS, satisfies the conditions 1], and produces only nonnegative solutions of problem (1), (2). Conditions (a) and (b) imply that the output v and the input u of Eq.…”

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“…Parabolic equations with gradient nonlinearities have been comprehensively studied [2] from various viewpoints. The first boundary value problem…”