WEAK AND YOUNG MEASURE SOLUTIONS FOR HYPERBOLIC INITIAL BOUNDARY VALUE PROBLEMS OF ELASTODYNAMICS IN THE ORLICZ-SOBOLEV SPACE SETTING.

We establish the existence result for a global weak solution (respectively, Young measure solution) in the Orlicz-Sobolev space setting for the nonlinear hyperbolic initial boundary value problem to utt = div(σ(Du)) + µ(Δu)t (respectively, utt = div(σ(Du))), where the function σ = @W=@F is continuous and the stored-energy function W : Md×n → R may be nonconvex. Our study is motivated by one-dimensional elastodynamics. The present paper gives first solvability results for nonlinear hyperbolic partial differential equations with nonpower-growth-type nonlinearity for Du in the monotonicity case and in the case with lack of monotonicity. The results are new even for the one-dimensional case with σ(Τ) = lnq(1 + |Τ|)|Τ|p-2Τ + aΤ for q > 0 and p ≥ 2 and a ∈ R; here a > 0 corresponds to the strong ellipticity of W, a = 0 -- the convexity of W, and a < 0 -- the nonconvexity of W under the Andrews-Ball inequality. [ABSTRACT FROM AUTHOR]