Finite-time blow-up for inhomogeneous parabolic equations with nonlinear memory. Issue 1 (2nd January 2021) - Alqahtani, Awatif
Abstract : In this paper, we study the effect of an inhomogeneity w = w ( x ) on the finite-time blow-up of solutions to the nonlinear heat equation ∂ t u − Δ u = 1 Γ ( 1 − γ ) ∫ 0 t ( t − s ) − γ | u ( s ) | p d s + w ( x ), ( t, x ) ∈ ( 0, T ) × R N, where N ≥ 1, 0 < γ < 1 and p >1. It is well known that in the homogeneous case w ≡ 0, the Fujita critical exponent is given by p ∗ = max 1 γ, 1 + 4 − 2 γ ( N − 2 + 2 γ ) + . In the case ∫ R N w ( x ) d x > 0 and u ( 0, ⋅ ) ≥ 0, we prove that the critical exponent is equal to ∞, which means that for all p >1, we have a finite-time blow-up.