Well-posedness of the Cauchy problem for the fractional power dissipative equation in critical Besov spaces. (English) Zbl 1151.35048
In this paper the Cauchy problem for the semilinear fractional power dissipative equation
\[
\begin{cases} u_t + (-\Delta)^\alpha u = F(u), & (t,x)\in \mathbb{R}^+ \times \mathbb{R}^n, \\ u(0,x)=u_0(x), & x\in\mathbb{R}^n, \end{cases}
\]
is studied, where \(\alpha>0\), \(b\in\mathbb{N}\), \(F(u)=P(D)u^{b+1}\) and \(P(D)\) is a homogeneous pseudo-differential operator of order \(d \in [0,2\alpha)\), and the initial data \(u_0\) belongs to the homogeneous Besov space \(\dot{B}^\sigma_{2,r}\) in the (critical) case \(\sigma=\frac{n}{2}-\frac{2\alpha-d}{b} > -\frac{n}{2}\), \(1\leq r\leq\infty\). The main result about the uniqueness and regularity of the solution is Theorem 1.1, complemented by the blow-up criterion Theorem 1.2. Proofs are presented in Section 4, preceded by estimates for the corresponding linear equation in the frame of mixed time-space setting, considered in Section 3. Furthermore, in addition to Fourier localization techniques and Littlewood-Paley assertions, a so-called “mono-norm method” is applied which differs from Kato’s “double-norm method” which was used in [C. Miao, B. Yuan, B. Zhang, Nonlinear Anal., Theory Methods Appl. 68, No. 3 (A), 461–484 (2008; Zbl 1132.35047)] for related assertions of well-posedness in Lebesgue spaces.
Reviewer: Dorothee Haroske (Jena)
MSC:
| 35K55 | Nonlinear parabolic equations |
| 35K15 | Initial value problems for second-order parabolic equations |
Citations:
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