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

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