## Taiwanese Journal of Mathematics

### ON SOME NONLINEAR DISSIPATIVE EQUATIONS WITH SUB-CRITICAL NONLINEARITIES

#### Abstract

We study the Cauchy problem for the nonlinear dissipative equations $$\left\{ \begin{array}{c} \partial _{t}u+\alpha \left( -\Delta \right) ^{\frac{\rho }{2}}u+\beta \left\vert u\right\vert ^{\sigma }u+\gamma \left\vert u\right\vert ^{\varkappa }u=0,\text{ }x\in {\mathbf{R}}^{n},t\gt 0, \\ u(0,x)=u_{0}(x),\text{ }x\in {\mathbf{R}}^{n}, \end{array} \right .$$ 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 $$\left\Vert u\left( t\right) \right\Vert _{\mathbf{L}^{\infty }}\leq \left\{ \begin{array}{c} C\left( 1+t\right) ^{-\frac{1}{\sigma }}\left( \log \left( 2+t\right) \right) ^{-\frac{1}{\sigma }}\text{ if }\sigma =\frac{\rho }{n}, \\ C\left( 1+t\right) ^{-\frac{1}{\sigma }}\text{ if }\sigma \in \left( 0,\frac{% \rho }{n}\right) . \end{array} \right.$$

#### Article information

Source
Taiwanese J. Math., Volume 8, Number 1 (2004), 135-154.

Dates
First available in Project Euclid: 20 July 2017

https://projecteuclid.org/euclid.twjm/1500558462

Digital Object Identifier
doi:10.11650/twjm/1500558462

Mathematical Reviews number (MathSciNet)
MR2058923

Zentralblatt MATH identifier
1052.35087

Subjects
Primary: 35Q35: PDEs in connection with fluid mechanics

#### Citation

Hayashi, Nakao; Ito, Naoko; Kaikina, Elena I.; Naumkin, Pavel I. ON SOME NONLINEAR DISSIPATIVE EQUATIONS WITH SUB-CRITICAL NONLINEARITIES. Taiwanese J. Math. 8 (2004), no. 1, 135--154. doi:10.11650/twjm/1500558462. https://projecteuclid.org/euclid.twjm/1500558462