On the spatial blow-up and decay for some nonlinear parabolic equations with nonlinear boundary conditions

Abstract

In this note we investigate the spatial behavior of several nonlinear parabolic equations with nonlinear boundary conditions. Under suitable conditions on the nonlinear terms we prove that the solutions either cease to exist for a finite value of the spatial variable or else they decay algebraically. The main tool used is the weighted energy method. Our results can be applied to several situations concerning heat conduction.

Some alternative results for some nonlinear parabolic equations in the half cylinder

“…In the present paper, we no longer suppose the condition on the nonlinear item rðx; u; q 2 Þ such that 0orrr M . In Sections 2 and 3, we choose r ¼ jruj p , we can also get the results similar to the results obtained in [24]. In Section 4, we choose r ¼ jruj 2 þ 1, we can obtain the traditional Phragm en-Lindelöf alternative results.…”

“…In [24], the author studied the spatial behavior of Eq. (1.1) when r 1, he proved that the smooth solution either fails to exists globally or when it does exist globally, it must tend asymptotically to zero with increasing long distance along the cylinder.…”

“…Papers dealing with ''blow-up'' are abound in the literature. The contributions concerning this question were explained in the work of Ames [1], the books of Ames and Straughan [2], Evans [6], Flavin and Rionero [7], and the work of Quintanilla [24], Payne and Schaefer [19,20] and Payne and Song [21]. Where the energy method is widely used to study the spatial behavior of solutions of partial differential equations.…”

On the asymptotic spatial behaviour of the solutions of the nerve system

“…Again the functions of the form f (v) = −a|v| n v + b|v| n 2 v + μ 1 v, where a, n and μ 1 are positive constants satisfy our requirements (see the Appendix for a proof). Our arguments in this section are inspired by the papers [8,9,22]. Another time the use of weighted functions allows us to overcome the mathematical difficulties of the analysis.…”

Some alternative results for the nonlinear viscoelasticity equations

“…The contributions concerning this question were explained in the [25][26][27][28][29]. Where the energy method is widely used to study the spatial behaviour of solutions of partial differential systems, but most of them were concerned with the elliptic or parabolic equations.…”