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).


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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[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 · doi:10.4310/PAMQ.2008.v4.n1.a2
[3] Cortissoz J.: Some elementary estimates for the Navier-Stokes system. Proc. Amer. Math. Soc. 137, 3343–3353 (2009) · Zbl 1176.35125 · doi:10.1090/S0002-9939-09-09989-4
[4] Dal Passo R., Luckhaus S.: A degenerate diffusion problem not in divergence form. J. Differential Equations 69, 1–14 (1987) · Zbl 0688.35045 · doi:10.1016/0022-0396(87)90099-4
[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 · doi:10.1090/conm/071/954419
[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 · doi:10.1007/s004400050135
[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 · doi:10.1142/S0219199799000183
[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 · doi:10.1006/jdeq.1998.3535
[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 · doi:10.1007/BF01769226
[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 · doi:10.1016/S0022-0396(03)00127-X
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