Blow-up and global existence for a general class of nonlocal nonlinear coupled wave equations

Abstract

We study the initial-value problem for a general class of nonlinear nonlocal coupled wave equations. The problem involves convolution operators with kernel functions whose Fourier transforms are nonnegative. Some well-known examples of nonlinear wave equations, such as coupled Boussinesq-type equations arising in elasticity and in quasi-continuum approximation of dense lattices, follow from the present model for suitable choices of the kernel functions. We establish local existence and sufficient conditions for finitetime blow-up and as well as global existence of solutions of the problem.

The relationships between some types of partial differential equations and ordinary differential equations as well as their applications

“…There are many literature on (5), we can refer to previous studies 12,34,35 and the references therein. Since we hope to obtain the bounds for the blowup time t * of u(x, t), we are only concerned with (5) in the case of p > 1.…”

“…There are many interesting topics on these problems, such as the conditions on global existence and blowup in finite time, estimates for the blowup rate, and blowup time of the solutions. By the results of previous works, [18][19][20][21][22][23][24][25] we know that the solution to (2) (or (1) or (5) will blow up in finite time under some assumptions; one of the essential conditions for (2) (or (1) or (5) is that the function f (or h) satisfies (x, ) ≥ c 1 > 0 (or h(x, )) with some p 1 > 1 for large and any x ∈ Ω. Yet we do not care about the conditions on the blowup in finite time and global existence of the solution in this paper; we focus on the lower and upper bounds for the blowup time of the solutions.…”

“…Since the equation has the nonlocal nonlinear term in each model, we call it nonlocal partial differential equation. There is an extensive literature on nonlocal parabolic equation or nonlocal wave equation, we can refer to previous studies [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] and the references therein.…”

“…Theorem 3. Assume that u is a nonnegative solution to (5), which becomes unbounded in L k+1 -norm at t = t * . Then a lower bound for blowup time of the solution is given by…”

Unidirectional wave motion in a nonlocally and nonlinearly elastic medium: the KdV, BBM, and CH equations

“…Recently, a nonlinear model of nonlocal elasticity was proposed in [5,6] in which the nonlocal stress is expressed as the weighted average of the local stress over space and consequently the nonlinear constitutive relation is based on a convolution integral over space. In subsequent papers [7,8], global existence and nonexistence of solutions of the initial-value problems posed for various generalizations of the model were investigated. In a very recent study [9], using an asymptotic expansion technique, the asymptotic equations governing unidirectional wave propagation of small-butfinite amplitude long waves in the nonlinear nonlocal elastic medium were derived.…”