On the existence of a global strong radial symmetric solution for a third-order nonlinear evolution equation in two space dimensions

Abstract

In this paper we consider the following nonlinear evolution equation: φ tt−Δφ− div 1 3 |∇φ| 2∇φ =Δφ t, x∈ R 2, t⩾0, with initial data φ( x,0)= φ 0( x), φ t ( x,0)= φ 1( x), φ 0∈H 3( R 2) , φ 1∈H 2( R 2) , which is an extension of a particular case of the unidimensional viscoelasticity equations. We prove the existence of a unique global strong solution φ∈C [0,+∞[;H 3 ∩C 1 [0,+∞[;H 2 ∩C 2 [0,+∞[;L 2 , without restrictions on the “size” of φ 0 and φ 1; if φ 0 and φ 1 only depend on r=| x|=( x 1 2+ x 2 2) 1/2, the solution φ only depends on r and t.