## 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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