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On the blow-up behavior of a nonlinear parabolic equation with periodic boundary conditions. (English) Zbl 1263.35047

The author considers the blow-up behavior of real solutions in the \(d\)-dimensional unit cube to the quasilinear problem \[ \left\{ \begin{alignedat}{2} &u_t=u\Delta u+u^2&&\qquad\text{on }[0,1]^d\times(0,T),\\ &u(\cdot,0)=\psi&&\qquad\text{in } [0,1]^d, \end{alignedat}\right.\tag{1} \] with periodic boundary conditions. Consider the family of seminorms for \(v\in L^2([0,1]^d)\) given by \[ \|v\|_{f(s),\beta} :=\sup_{\xi\in\mathbb{Z}^d,\xi\neq0} |\beta|^\beta|f(\xi)|\,|\hat v(\xi)|, \] where \(\hat v\) denotes the Fourier transform of \(v\). The following result is proved: put \(\alpha(1):=3/2\) and \(\alpha(d):=d\) if \(d\geq 2\). There are positive constants \(c_d\) and \(C_k\), \(k\in\mathbb{N}\), such that if \(\psi\in L^2([0,1]^d)\) satisfies \[ \int_{[0,1]^d}\psi \geq c_d\|\psi\|_{\log^{\frac32}(|s|+2),\alpha(d)}, \] then the Cauchy problem (1) has a unique solution \(u\) with initial condition \(\psi\). This solution blows up at a time \(T\in(0,\infty)\) and satisfies \[ \biggl\|u(x,t)-\int_{[0,1]^d}u(x,t)\,dx\biggr\|_{C^k([0,1]^d)} <C_k(T-t)\qquad\text{for all }t\in[0,T),\;k\in\mathbb{N}. \] In other words, its limiting profile is flat.
The proof rests on Fourier transforms and Galerkin approximations for solutions of (1).

MSC:

35B44 Blow-up in context of PDEs
35K59 Quasilinear parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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