Critical exponent for damped wave equations with nonlinear memory

Abstract

We consider the Cauchy problem in R n , n ≥ 1, for a semilinear damped wave equation with nonlinear memory. Global existence and asymptotic behavior as t → ∞ of small data solutions have been established in the case when 1 ≤ n ≤ 3. Moreover, we derive a blow-up result under some positive data in any dimensional space.

Existence and blow-up of solutions for damped ave system with nonlinear memory

“…(i) Theorem 2.3 is sharp in the case (6) (respectively, (10)), see (4)- (14) and generalizes Theorem 2 of [11]. Indeed let p = q, γ 1 = γ 1 = γ .…”

“…Remark 2 By using the same techniques as in [11], we can obtain the well-posedness for any dimension space N ≥ 1, as follows: Let 1 < p, q ≤ N / (N − 2) for N ≥ 3, and Downloaded by [University of Nebraska, Lincoln] at 23:48 03 January 2015 (1) possesses a unique maximal mild solution (u, v) satisfies the integral system (27), such that…”

“…The main tool used in [11] is the weighted energy method with a weight similar to the one introduced in [9]. It was shown that the solution of (3) behaves as that of the corresponding diffusive Equation (2).…”

“…The method used in [14] is inspired from the weighted energy method developed by Todorova and Yordanov [9] and used recently by Fino [11]. As we have seen, Yongqiang restricts himself in the case of compactly supported data and the dimension N = 1.…”

“…Motivated by the papers [10,11,14,15], in the present paper, we consider the problem (1), we will give conditions relating the space dimension N with the system of parameters γ 1 ;γ 2 ; p and q for which the solution exists globally in time and satisfies L ∞ -decay estimates, as well as a blow-up in finite time of solution with initial data having positive average.…”