Blow-up of solutions to parabolic equations with nonstandard growth conditions

Abstract

We study the phenomenon of finite time blow-up in solutions of the homogeneous Dirichlet problem for the parabolic equation u t = div ( a ( x , t ) | ∇ u | p ( x ) − 2 ∇ u ) + b ( x , t ) | u | σ ( x , t ) − 2 u with variable exponents of nonlinearity p ( x ) , σ ( x , t ) ∈ ( 1 , ∞ ) . Two different cases are studied. In the case of semilinear equation with p ( x ) ≡ 2 , a ( x , t ) ≡ 1 , b ( x , t ) ≥ b − > 0 we show that the finite time blow-up happens if the initial function is sufficiently large and either min Ω σ ( x , t ) = σ − ( t ) > 2 for all t > 0 , or σ − ( t ) ≥ 2 , σ − ( t ) ↘ 2 as t → ∞ and ∫ 1 ∞ e s ( 2 − σ − ( s ) ) d s < ∞ . In the case of the evolution p ( x ) -Laplace equation with the exponents p ( x ) , σ ( x ) independent of t , we prove that every solution corresponding to a sufficiently large initial function exhibits a finite time blow-up if b ( x , t ) ≥ b − > 0 , a t ( x , t ) ≤ 0 , b t ( x , t ) ≥ 0 , min σ ( x ) > 2 and max p ( x ) ≤ min σ ( x ) .