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Asymptotic for fractional nonlinear heat equations. (English) Zbl 1183.35037

Summary: The Cauchy problem is studied for the nonlinear equations with fractional power of the negative Laplacian
\[ \begin{cases} u_t+(-\Delta)^{\alpha/2}u+ u^{1+\sigma}=0, &x\in\mathbb R^n,\;t>0,\\ u(0,x)= u_0(x), &x\in\mathbb R^n, \end{cases} \]
where \(\alpha\in (0,2)\), with critical \(\sigma=\alpha/n\) and sub-critical \(\sigma\in (0,\alpha/n)\) powers of the nonlinearity. Let \(u_0\in{\mathbf L}^{1,a}\cap {\mathbf L}^\infty\cap\mathbb C\), \(u_0(x)\geq 0\) in \(\mathbb R^n\), \(\theta= \int_{\mathbb R^n}u_0(x)\,dx>0\). The case of not small initial data is of interest. It is proved that the Cauchy problem has a unique global solution \(u\in\mathbb C([0,\infty); {\mathbf L}^\infty\cap {\mathbf L}^{1,a}\cap\mathbb C)\) and the large time asymptotics are obtained.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations
35K15 Initial value problems for second-order parabolic equations
35R11 Fractional partial differential equations
35B33 Critical exponents in context of PDEs
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