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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References:

 [1] Angenent S.: On the formation of singularities in the curve shortening flow. J. Differential Geom. 33, 601–633 (1991) · Zbl 0731.53002 [2] 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 [3] Cortissoz J.: Some elementary estimates for the Navier-Stokes system. Proc. Amer. Math. Soc. 137, 3343–3353 (2009) · Zbl 1176.35125 [4] Dal Passo R., Luckhaus S.: A degenerate diffusion problem not in divergence form. J. Differential Equations 69, 1–14 (1987) · Zbl 0688.35045 [5] Friedman A., McLeod B.: Blow-up of solutions of nonlinear parabolic equations. Arch. Rational Mech. Anal. 96, 55–80 (1987) · Zbl 0619.35060 [6] Gage M., Hamilton R.S.: The heat equation shrinking convex plane curves. J. Differential Geom. 23, 69–96 (1986) · Zbl 0621.53001 [7] Hamilton R.S.: The Ricci flow on surfaces, Mathematics and General Relativity. Contemporary Mathematics 71, 237–261 (1988) · Zbl 0663.53031 [8] 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 [9] 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 [10] 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 [11] Ughi M.: A degenerate parabolic equation modelling the spread of an epidemic. Ann. Mat. Pura Appl. (4) 143, 385–400 (1986) · Zbl 0617.35066 [12] 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
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