Global well-posedness for nonlinear nonlocal Cauchy problems arising in elasticity

Abstract

<p>In this article, we prove global well-posedness for a family of one dimensional nonlinear nonlocal Cauchy problems arising in elasticity. We consider the equation</p> <pre>u_tt - δLu_xx = (β ⁎ [(1 - δ)u + u²ⁿ⁺¹])_xx,</pre> <p>where L is a differential operator, β is an integral operator, and δ = 0 or 1. (Here, the case δ = 1 represents the additional doubly dispersive effect.) We prove the global well-posedness of the equation in energy spaces.</p>