## 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
Full Text:

### References:

  Angenent S.: On the formation of singularities in the curve shortening flow. J. Differential Geom. 33, 601–633 (1991) · Zbl 0731.53002  Arnold M.D., Sinai Ya.G.: Global Existence and Uniqueness Theorem for 3D-Navier Stokes System on $${$$\backslash$$mathbb{T}\^3}$$ for small initial conditions in the spaces {$$\Phi$$}({$$\alpha$$}). Pure Appl. Math. Q. 4, 71–79 (2008) · Zbl 1146.35074  Cortissoz J.: Some elementary estimates for the Navier-Stokes system. Proc. Amer. Math. Soc. 137, 3343–3353 (2009) · Zbl 1176.35125  Dal Passo R., Luckhaus S.: A degenerate diffusion problem not in divergence form. J. Differential Equations 69, 1–14 (1987) · Zbl 0688.35045  Friedman A., McLeod B.: Blow-up of solutions of nonlinear parabolic equations. Arch. Rational Mech. Anal. 96, 55–80 (1987) · Zbl 0619.35060  Gage M., Hamilton R.S.: The heat equation shrinking convex plane curves. J. Differential Geom. 23, 69–96 (1986) · Zbl 0621.53001  Hamilton R.S.: The Ricci flow on surfaces, Mathematics and General Relativity. Contemporary Mathematics 71, 237–261 (1988) · Zbl 0663.53031  Le Jan Y., Sznitman A.S.: Stochastic cascades and 3-dimensional Navier-Stokes equations. Probab. Theory Related Fields 109, 343–366 (1997) · Zbl 0888.60072  Mattingly J., Sinai Ya.G.: An elementary proof of the existence and uniqueness theorem for the Navier Stokes equation. Commun. Contemp. Math. 1, 497–516 (1999) · Zbl 0961.35112  Souplet P.: Uniform Blow Up and Boundary Behavior for Diffusion Equations with Nonlocal Nonlinear Source. J. Diff. Equations 153, 374–406 (1999) · Zbl 0923.35077  Ughi M.: A degenerate parabolic equation modelling the spread of an epidemic. Ann. Mat. Pura Appl. (4) 143, 385–400 (1986) · Zbl 0617.35066  Winkler M.: Blow-up of solutions to a degenerate parabolic equation not in divergence form. J. Diff. Equations 192, 445–474 (2003) · Zbl 1028.35081
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.