Combined effects of two nonlinearities in lifespan of small solutions to semi-linear wave equations

Abstract

This paper investigates the combined effects of two distinctive power-type nonlinear terms (with parameters p, q > 1) in the lifespan of small solutions to semilinear wave equations. We determine the full region of ( p, q) to admit global existence of small solutions, at least for spatial dimensions n = 2, 3. Moreover, for many ( p, q) when there is no global existence, we obtain sharp lower bound of the lifespan, which is of the same order as the upper bound of the lifespan. Mathematics Subject ClassificationIn memory of Rentaro Agemi.The authors are very grateful to the referees for their helpful comments.

Blow-up phenomena of semilinear wave equations and their weakly coupled systems

“…for some a, b > 0 and p, q > 1. This problem has been considered by Zhou-Han [64] Hidano-Wang-Yokoyama [19] and Wang-Zhou [56]. In this case the definition of weak solutions is the following:…”

“…Remark 6.1. In the case Γ S (N, q) < 0, Γ G (N, p) < 0 and Γ comb (N, p, q) ≤ 0, Hidano-Wang-Yokoyama [19] proved global existence of small solutions to (6.1) when N = 2, 3. Therefore although it is open but one can expect that the same conclusion can be proved for all dimensions.…”

“…We should remark that the global existence of small solutions to (1.1) with G = |∂ t u| p is only proved under the initial value with radially symmetric in high spatial dimension. The problem (1.1) with combined type G = |u| q + |∂ t u| p has been recently discussed by Zhou-Han [64] and Hidano-Wang-Yokoyama [19]. In [19], it is found the borderline of the position of (p, q) for blowup phenomena of small solutions.…”

Global existence and lifespan for semilinear wave equations with mixed nonlinear terms

“…Firstly we study the lifespan of the equation with mixed nonlinear terms (1.1) u := ∂ 2 t u − ∆u = |∂ t u| p + |u| q , (u, ∂ t u)| t=0 = ε f (x), g(x) , where p > 1, q > 1 and x ∈ R n . This equation is in relation ( [9]) with the following equations which are well-investigated: v = |v| q , t > 0, x ∈ R n , (1.2) w = |∂ t w| p , t > 0, x ∈ R n . (1.…”

“…For example, the obtained bound exp cε −qc(qc−1) which comes from [14] coincides with q = q c in estimates (3.1), and the obtained bound exp cε −(qc−1) 2 /2 which comes from [11] coincides with q = (q c − 1)/2 in estimates (3.2). In order to improve the result from [9], we adapt these generalized Strichartz estimates to the equation (1.1), use energy inequality with Klainerman-Sobolev inequality to deal with derivative term. Thus we get the following result for dimension three, which is sharp in general.…”