The Initial-Boundary Value Problem For A Kind Of Nonlinear Fourth-Order Wave Equations

For the initial-boundary value problem of a kind of nonlinear fourth-order wave equations:where Rn is bounded domain with sufficiently smooth boundary.What studied in this paper are the global existence, uniqueness, smoothness and blow-up of the weak solutions of (0.1) - (0.3). Our four main results are stated as follows:1 By using the Galerkin method and constructing stable setaccording to the potential well theory, It is proved:Theorem (existence): Let . Then problem (0.1)-(0.3) has global weak solutions u satisfying:2 If p satisfies appropriately stronger condi tions, by using the energy method and the trick of inequality, It is proved:Theorem (uniqueness): Let . Then the global weak solutionof problem (0.1)-(0.3) is unique.3 By using the Galerkin method, stable set and trick of inequality, It is proved:Theorem (smoothness) : Let . Then the problem (0.1)-(0.3) has unique global weak solution u satisfying:4 By the convexity method and constructing unstable set according to the potential well theory, It is proved:Theorem (blow up): Let u, u is thelocal solution of problem (0.1) ?0.3), The there exists a finite constant T, such that. u blows up in f ini tetime under the L2() norm.