Abstract
This paper deals with the initial boundary value problem for the nonlinear beam equation with double damping terms \t\t\tutt−uxxt+uxxxx+uxxxxt=g(uxx)xx,x∈Ω,t>0,\\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} $$\\begin{aligned} u_{tt}-u_{xxt}+u_{xxxx}+u_{xxxxt}=g(u_{xx})_{xx}, \\quad x\\in\\Omega, t>0, \\end{aligned}$$ \\end{document} where Omega=(0,1), and g(s) is a given nonlinear function. We derive sufficient conditions for the blow-up of the solution to the problem by virtue of an adapted concavity method. In addition, global existence of weak solutions as well as exponential and uniform decay rates of the solution energy are established by the use of an integral inequality.
Highlights
1 Introduction In this paper, we study the following initial boundary value problem: utt – uxxt + uxxxx + uxxxxt = g(uxx)xx, x ∈, t > 0, u(0, t) = u(1, t) = 0, ux(0, t) = ux(1, t) = 0, t ≥ 0, u(x, 0) = u0(x) ∈ H02( ), ut(x, 0) = u1(x) ∈ L2( ), (1.1) (1.2) (1.3)
In [5], the existence and uniqueness of the generalized global solutions and the classical global solutions of the three different initial boundary value problems for a damped nonlinear hyperbolic equation are proved by Chen
We focus on finding sufficient conditions yielding nonexistence of a global solution to the above problem
Summary
The global existence, uniqueness, regularity and continuous dependence on the initial data of a generalized solution to the problem (1.4)–(1.6) are proved under general conditions on the nonlinear term. This class contains problem (1.1)–(1.3) as a particular example. In [3], the existence and uniqueness of weak solutions to a class of nonlinear beam equation are established under certain assumptions (locally Lipschitz plus affine domination) on the nonlinearity. Their results weaken the stringent monotonicity assumptions in the previous theories. Sufficient conditions for a blow-up of solutions are given in [5]
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