On some nonlinear dispersive equations in several space variables (Q1174373)

The authors treat nonlinear dispersive systems of the form \[ u_ t- \Delta u_ t=\text{div}f(u),\qquad (x,t)\in\Omega\times(0,\infty) \leqno(1) \] subject to the Dirichlet boundary condition and the initial condition \(u(x,0)=u_ 0(x)\). They show that if \(u_ 0\in W^{2,p}(\Omega)\cap W_ 0^{1,p}(\Omega)\) for some \(p\) with \(\max\{n/2,1\}<p<\infty\), then the problem (1) has a unique strong solution \(u\) in \(C^ 1([0,T);W^{2,p}(\Omega)\cap W_ 0^{1,p}(\Omega))\) on some time interval \([0,T)\). Some other existence theorems for strong or weak solutions are also proved.