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Well-posedness of the Cauchy problem for the fractional power dissipative equations. (English) Zbl 1132.35047
Of concern in this paper is the following Cauchy problem
\begin{aligned} u_{t}+(-\Delta)^{\alpha}u=F(u),\;(t,x)\in \mathbb{R}^{+}\times \mathbb{R} ^{n}, \\ u(0,x)=\varphi (x),\;x\in \mathbb{R}^{n}, \end{aligned} \tag{1} with the nonlinear term $$F(u)$$ is equal either to $$f(u)$$ or to $$Q(D)f(u),$$ where $$Q(D)$$ is a homogeneous pseudo-differential operator of order $$d\in [0,2\alpha )$$ with real number $$\alpha >0$$ and $$f(u)$$ is a function which behaves like $$\left| u\right| ^{b}u$$ or $$\left| u\right| ^{b_{1}}u+\left| u\right| ^{b_{2}}u$$ with $$b,b_{1},b_{2}>0.$$ The family of equations in this problem covers many well known equations such as the semilinear fractional power dissipative equations, the dissipative quasi-geostrophic equation, the generalized convection-diffusion equation and the Ginzburg-Landau equation. The authors present a unified method to deal with the well-posedness for (1) with initial data in $$L^{r}(\mathbb{R}^{n})$$ ($$r\geq r_{0}:=nb/(2\alpha -d)$$ ) or in the Besov space $$\mathbb{\dot{B}}_{p,\infty }^{-\sigma }(\mathbb{R} ^{n})$$ ($$\sigma :=(2\alpha -d)/b-np$$ and $$1\leq p\leq \infty )$$ in appropriate space-time spaces such as $$C([0,\infty );L^{r}(\mathbb{R} ^{n}))\cap L^{q}([0,\infty );L^{p}(\mathbb{R}^{n}))$$ or $$C(I;L^{r}(\mathbb{R} ^{n}))\cap C_{q}(I;L^{p}(\mathbb{R}^{n})).$$ Many interesting results are proved.

MSC:
 35K55 Nonlinear parabolic equations 35K15 Initial value problems for second-order parabolic equations 26A33 Fractional derivatives and integrals
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References:
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