Abstract
This paper focuses on the initial boundary value problem of semilinear wave equation in exterior domain in two space dimensions with critical power. Based on the contradiction argument, we prove that the solution will blow up in a finite time. This complements the existence result of supercritical case by Smith, Sogge and Wang [ 20 ] and blow up result of subcritical case by Li and Wang [ 14 ] in two space dimensions.
Highlights
We consider the initial boundary value problem of the semilinear wave equation with Dirichlet boundary condition utt − ∆u = |u|p, t > 0, x ∈ B1c= R2 \ B1(O), u(0, x) = εf (x), ut(0, x) = εg(x), x ∈ B1c = R2 \ B1(O), (1)u|∂B1(O) = 0, where p = pc(2) is the positive root of the following quadratic equation p2 − 3p − 2 = 0. (2)And B1(O) denotes the unit ball centered at the origin in R2, ε represents the smallness of the data.2000 Mathematics Subject Classification. 35L05, 35L70
For semilinear wave equations with small data, we often come to Strauss conjecture, which concerns the global existence and blow up of solutions to the following
To solve the critical initial boundary value problem for n ≥ 5 heavily relies on the fast decay rate of a special solution of the linear wave equation
Summary
Critical semilinear wave equations, initial boundary value problem. ∗ Corresponding author: Ning-An Lai. For semilinear wave equations with small data, we often come to Strauss conjecture, which concerns the global existence and blow up of solutions to the following. We aim to establish blow up result to initial boundary value problem (1) outside a unit ball. To solve the critical initial boundary value problem for n ≥ 5 heavily relies on the fast decay rate of a special solution of the linear wave equation. It is easy to see that they have different asymptotic behavior Another key ingredient is that we use a special test function φq which is positive, homogeneous of degree q(q > 0) and radially symmetric.
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