Cauchy problem for non-linear systems of equations in the critical case

E. I. Kaikina ; P. I. Naumkin ; I. A. Shishmarev

Sbornik. Mathematics, Tome 195 (2004) no. 11, pp. 1575-1605

Résumé

The large-time asymptotic behaviour is studied for a system of non-linear evolution dissipative equations \begin{gather*} u_t+\mathscr N(u,u)+\mathscr Lu=0, \qquad x\in\mathbb R^n, \quad t>0, \\ u(0,x)=\widetilde u(x), \qquad x\in\mathbb R^n, \end{gather*} where $\mathscr L$ is a linear pseudodifferential operator $\mathscr Lu=\overline{\mathscr F}_{\xi\to x}(L(\xi)\widehat u(\xi))$ and the non-linearity $\mathscr N$ is a quadratic pseudodifferential operator $$ \mathscr N(u,u)=\overline{\mathscr F}_{\xi\to x}\sum_{k,l=1}^m\int_{\mathbb R^n}A^{kl}(t,\xi,y)\widehat u_k(t,\xi-y)\widehat u_l(t,y)\,dy, $$ where $\widehat u\equiv\mathscr F_{x\to\xi}u$ is the Fourier transform. Under the assumptions that the initial data $\widetilde u\in\mathbf H^{\beta,0}\cap\mathbf H^{0,\beta}$, $\beta>n/2$ are sufficiently small, where $$ \mathbf H^{n,m}=\{\phi\in\mathbf L^2:\|\langle x\rangle^m\langle i\partial_x\rangle^n\phi(x)\|_{\mathbf L^2}\infty\}, \qquad \langle x\rangle=\sqrt{1+x^2}\,, $$ is a Sobolev weighted space, and that the total mass vector $\displaystyle M=\int\widetilde u(x)\,dx\ne0$ is non-zero it is proved that the leading term in the large-time asymptotic expansion of solutions in the critical case is a self-similar solution defined uniquely by the total mass vector $M$ of the initial data.