ON SOME NONLINEAR DISSIPATIVE EQUATIONS WITH SUB-CRITICAL NONLINEARITIES

Abstract

We study the Cauchy problem for the nonlinear dissipative equations \begin{equation} \left\{ \begin{array}{c} \partial _{t}u+\alpha \left( -\Delta \right) ^{\frac{\rho }{2}}u+\beta \left\vert u\right\vert ^{\sigma }u+\gamma \left\vert u\right\vert ^{\varkappa }u=0,\text{ }x\in {\mathbf{R}}^{n},t\gt 0, \\ u(0,x)=u_{0}(x),\text{ }x\in {\mathbf{R}}^{n}, \end{array} \right . \end{equation} where $\alpha ,\beta ,\gamma \in {\mathbf{C}},$ Re $\alpha \gt 0$, $\rho \gt 0$, $\varkappa \gt \sigma \gt 0.$ We are interested in the critical case, $\sigma =% \frac{\rho }{n}$ and sub critical cases $0\lt \sigma \lt \frac{\rho }{n}$. We assume that the initial data $u_{0}$ are sufficiently small in a suiatble norm, $\left\vert \int u_{0}\left( x\right) dx\right\vert \gt 0$ and Re$\beta \delta (\alpha ,\rho ,\sigma )\gt 0$, where $$ \delta (\alpha ,\rho ,\sigma )=\int \left\vert G\left( x\right) \right\vert ^{\sigma }G\left( x\right) dx $$ and $G\left( x\right) ={\mathcal{F}}^{-1}e^{-\alpha \left\vert \xi \right\vert ^{\rho }}.$ In the sub critical case we assume that $\sigma $ is close to $\frac{\rho }{n}.$ Then we prove global existence in time of solutions to the Cauchy problem (1) satisfying the time decay estimate $$ \left\Vert u\left( t\right) \right\Vert _{\mathbf{L}^{\infty }}\leq \left\{ \begin{array}{c} C\left( 1+t\right) ^{-\frac{1}{\sigma }}\left( \log \left( 2+t\right) \right) ^{-\frac{1}{\sigma }}\text{ if }\sigma =\frac{\rho }{n}, \\ C\left( 1+t\right) ^{-\frac{1}{\sigma }}\text{ if }\sigma \in \left( 0,\frac{% \rho }{n}\right) . \end{array} \right. $$