Blow-up for parabolic and hyperbolic problems with variable exponents

Abstract In this paper, we study the blow up problem for positive solutions of parabolic and hyperbolic problems with reaction terms of local and nonlocal type involving a variable exponent. We prove the existence of initial data such that the corresponding solutions blow up at a finite time.

[1]  S. Hofmann.  Weighted norm inequalities and vector valued inequalities for certain rough operators , 1993 .

[2]  Zhou Gang,et al.  Blow-up in nonlinear heat equations , 2008, Adv. Appl. Math..

[3]  Chia-Ven Pao,et al.  Nonlinear parabolic and elliptic equations , 1993 .

[4]  M. A. Herrero,et al.  Generic behaviour of one-dimensional blow up patterns , 1992 .

[5]  Michel Chipot,et al.  Stationary partial differential equations , 2004 .

[6]  V. Zhikov,et al.  AVERAGING OF FUNCTIONALS OF THE CALCULUS OF VARIATIONS AND ELASTICITY THEORY , 1987 .

[7]  Boris Hasselblatt,et al.  Handbook of Dynamical Systems , 2010 .

[8]  T C Halsey.  Electrorheological fluids. , 1992, Science.

[9]  Yoshikazu Giga,et al.  Characterizing Blow-up Using Similarity Variables , 1985 .

[10]  The blow-up problem for a semilinear parabolic equation with a potential , 2006, math/0607055.

[11]  Sergey Shmarev,et al.  Chapter 1 Elliptic equations with anisotropic nonlinearity and nonstandard growth conditions , 2006 .

[12]  S. Antontsev,et al.  Parabolic Equations with Anisotropic Nonstandard Growth Conditions , 2006 .

[13]  M. Ruzicka,et al.  Electrorheological Fluids: Modeling and Mathematical Theory , 2000 .

[14]  Hiroshi Matano,et al.  Chapter 14 – Blow-up in Nonlinear Heat Equations from the Dynamical Systems Point of View , 2002 .

[15]  Philippe Souplet,et al.  Blow-up in nonlocal reaction-diffusion equations , 1998 .

[16]  R. Glassey.  Blow-up theorems for nonlinear wave equations , 1973 .

[17]  Sergey Shmarev,et al.  Existence and uniqueness of solutions of degenerate parabolic equations with variable exponents of nonlinearity , 2008 .

[18]  Miguel A. Herrero,et al.  Blow-up behaviour of one-dimensional semilinear parabolic equations , 1993 .

[19]  A. P. Mikhailov,et al.  Blow-Up in Quasilinear Parabolic Equations , 1995 .