Solution asymptotics for nonlinear dissipative equations at large \(x\) and \(t\) (Q1200394)

The author studies the asymptotic behaviour for large \(x,t\) of solutions \(u(x,t)\) of the model equation \[ u_ t + \left( {\partial^ su \over \partial x^ s} \right)^ 2 + K(u) = 0,\;s=0,1, \quad K(u) = {1 \over 2 \pi} \int^ \infty_{-\infty} \exp (ipx) k(p) \widehat u(p,t)dp \] where \(\widehat u\) denotes the Fourier transform with respect to \(x\). The abstract equation includes some important special cases as e.g. the Burgers or Korteweg-de Vries equations. Two theorems are reported giving growth estimates of \(u\) and its partial derivatives when \(x,t \to \infty\). The paper contains no proofs.