Concerning the Strauss Conjecture for the subcritical and critical cases on the exterior domain in two space dimensions

The aim of this paper is to verify the Strauss Conjecture for semilinear wave equations with variable coefficients and for the subcritical and critical cases on the exterior domain for when the number of space dimensions n=2n=2. The main ingredients include generalized Sideris ODE blow-up results, and the existence and a priori estimates of the test functions (see Section 2). For this purpose, we study the following initial–boundary value problem for two-dimensional semilinear wave equations with variable coefficients on the exterior domain with subcritical and critical exponents: equation(0.1)utt−∑i,j=12∂i(aij(x)∂ju)=|u|p,(x,t)∈Ωc×(0,+∞),n=2, where u=u(x,t)u=u(x,t) is a real-valued scalar unknown function in Ωc×[0,+∞)Ωc×[0,+∞), where ΩΩ is a smooth compact obstacle in R2R2 and ΩcΩc is its complement.Two-dimensional initial–boundary value problems for semilinear wave equations with variable coefficients on the exterior domain are difficult to deal with because the fundamental solution of the harmonic equation is different from that for the higher dimensional case, and thus there will be a growth factor in the logarithmic term in the key test function ϕ0ϕ0 (see Section 2).One of the key points of our paper is the improvement of a well-known ODE blow-up result: Lemma 2.1 of [12, p. 386], for when the differential inequality involves a logarithmic term; and we obtain an estimate of the upper bound for the blow-up time. On the basis of the key results Lemma 2.3 and Lemma 2.4 obtained, we will establish a blow-up result for the above initial–boundary value problem; it is proved that there can be no global solutions, no matter how small the initial data are, and a lifespan estimate for the solutions for this problem will also be given.