ASYMPTOTICS FOR FRACTIONAL NONLINEAR HEAT EQUATIONS

Abstract

The Cauchy problem is studied for the nonlinear equations with fractional power of the negative Laplacian \[ \left\{ \begin{array}{@{}r@{\,}c@{\,}l@{\qquad}l} u_{t}+( -\Delta ) ^{{\alpha}/{2}}u+u^{1+\sigma } &=&0, & x\in {\mathbf{R}}^{n},\text{ }t>0, \\[4pt] u( 0,x) &=& u_{0} ( x), &x\in {\mathbf{R}}^{n}, \end{array} \right. \label{A} \] where $\alpha \in ( 0,2)$, with critical $\sigma ={\alpha }/{ n}$ and sub-critical $\sigma \in ( 0,{\alpha }/{n}) $ powers of the nonlinearity. Let $u_{0}\,{\in}\, \mathbf{L}^{1,a}\cap \mathbf{L}^{\infty }\cap \mathbf{C}, $$u_{0}( x) \,{\geq}\, 0$ in $\mathbf{R}^{n},$$\theta \,{=}\,\int_{ \mathbf{R}^{n}}u_{0}( x) \,dx\,{>}\,0.$ The case of not small initial data is of interest. It is proved that the Cauchy problem has a unique global solution $u \in \mathbf{C}( [0,\infty); \mathbf{L}^{\infty }\cap \mathbf{L}^{1,a}\cap \mathbf{C})$ and the large time asymptotics are obtained.