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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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