Abstract
In this paper, we study the initial boundary value problem for a Petrovsky type equation with a memory term, nonlinear weak damping, and a superlinear source: \t\t\tutt+Δ2u−∫0tg(t−τ)Δ2u(τ)dτ+|ut|m−2ut=|u|p−2u,in Ω×(0,T).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ u_{tt}+\\Delta ^{2} u- \\int _{0}^{t} g(t-\\tau )\\Delta ^{2} u(\\tau )\\,\\mathrm{d} \\tau + \\vert u_{t} \\vert ^{m-2}u_{t}= \\vert u \\vert ^{p-2}u,\\quad \\text{in }\\varOmega \\times (0,T). $$\\end{document} When the source is stronger than dissipations, we obtain the existence of certain weak solutions which blow up in finite time with initial energy E(0)=R for any given Rgeq 0.
Highlights
1 Introduction In this paper, we are concerned with the following initial boundary value problem for a nonlinear Petrovsky type equation with a memory term and nonlinear damping:
In [14], Chen and Zhou studied problem (1.2) and obtained the finite time blow-up results provided that the initial energy is positive bounded or vanishing
By using a special differential inequality, the author obtained the existence of certain solutions with nonpositive, positive bounded, or arbitrary high initial energy that blow up in finite time and derived the life span estimates of these solutions
Summary
In [14], Chen and Zhou studied problem (1.2) and obtained the finite time blow-up results provided that the initial energy is positive bounded or vanishing. The authors proved that, for any given α > 0, λ ≥ 0, there exist infinitely many data u0, u1 in the energy space such that the initial energy E(0) = λ, the gradient norm ∇u0 L2 = α, and the solution of the Cauchy problem blows up in finite time. By using a special differential inequality, the author obtained the existence of certain solutions with nonpositive, positive bounded, or arbitrary high initial energy that blow up in finite time and derived the life span estimates of these solutions.
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