Well-posedness of the supercritical Lane–Emden heat flow in Morrey spaces

Blatt, Simon¹ ; Struwe, Michael² 

¹ Fachbereich Mathematik, Universität Salzburg, Hellbrunner Str. 34, 5020 Salzburg, Austria
² Departement Mathematik, ETH-Zürich, 8092 Zürich, Switzerland

ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 4, pp. 1370-1381.

Résumé

For any smoothly bounded domain $\Omega \subset {ℝ}^n$, $n\ge 3$, and any exponent $p>2^{*}=2n/(n-2)$ we study the Lane–Emden heat flow $u_t-\Delta u={\left|u\right|}^{p-2}u$ on $\Omega \times ]0,T[$ and establish local and global well-posedness results for the initial value problem with suitably small initial data ${u|}_{t=0}=u_0$ in the Morrey space $L^{2,\lambda }\left(\Omega \right)$ for suitable $T\le \infty$, where $\lambda =4/(p-2)$. We contrast our results with results on instantaneous complete blow-up of the flow for certain large data in this space, similar to ill-posedness results of Galaktionov–Vazquez for the Lane–Emden flow on ${ℝ}^n$.

Détail

DOI : 10.1051/cocv/2016041

Classification : 35K55
Keywords: Nonlinear parabolic equations, well-posedness of initial-boundary value problem

Bibliographie

D.R. Adams, A note on Riesz potentials. Duke Math. J. 42 (1975) 765–778. | Zbl | MR | DOI

J.M. Ball, Finite time blow-up in nonlinear problems. Nonlinear evolution equations. Proc. of Sympos., Univ. Wisconsin, Madison, Wis., 1977. Academic Press, New York-London (1978) 189–205. | Zbl | MR

P. Baras and L. Cohen, Complete blow-up after $T_{max}$ for the solution of a semilinear heat equation. J. Funct. Anal. 71 (1987) 142–174. | Zbl | MR | DOI

S. Blatt and M. Struwe, An analytic framework for the supercritical Lane–Emden equation and its gradient flow. Int. Math. Res. Notices 2015 (2015) 2342–2385. | Zbl | MR

S. Blatt and M. Struwe, Boundary regularity for the supercritical Lane–Emden heat flow. Calc. Var. 54 (2015) 2269–2284. Publisher’s erratum. Calc. Var. 54 (2015) 2285. | Zbl | MR | DOI

H. Brezis and T. Cazenave, A nonlinear heat equation with singular initial data. J. Anal. Math. 68 (1996) 277–304. | Zbl | MR | DOI

A. Friedman, Partial differential equations. Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London (1969). | Zbl | MR

H. Fujita, On the blowing up of solutions of the Cauchy Problem for $u_t=\Delta u+u^{1+\alpha }$. J. Fac. Sci. Univ. Tokyo Sect. I 13 (1996) 109–124. | Zbl | MR

V.A. Galaktionov and J.L. Vazquez, Continuation of blowup solutions of nonlinear heat equations in several space dimensions. Commun. Pure Appl. Math. 50 (1997) 1–67. | Zbl | MR | 3.0.CO;2-H class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI

D.D. Joseph and T.S. Lundgren, Quasilinear Dirichlet problems driven by positive sources. Arch. Rational Mech. Anal. 49 (1972/73) 241–269. | Zbl | MR | DOI

S. Kaplan, On the growth of solutions of quasi-linear parabolic equations. Commun. Pure. Appl. Math. 16 (1963) 305–330. | Zbl | MR | DOI

H. Koch and D. Tataru, Well-posedness for the Navier–Stokes equations. Adv. Math. 157 (2001) 22–35. | Zbl | MR | DOI

T. Lamm, F. Robert and M. Struwe, The heat flow with a critical exponential nonlinearity. J. Funct. Anal. 257 (2009) 2951–2998. | Zbl | MR | DOI

H. Matano and F. Merle, Classification of type I and type II behaviors for a supercritical nonlinear heat equation. J. Funct. Anal. 256 (2009) 992–1064. | Zbl | MR | DOI

M.E. Taylor, Analysis on Morrey spaces and applications to Navier–Stokes and other evolution equations. Commun. Partial Differ. Equ. 17 (1992) 1407–1456. | Zbl | MR | DOI

F.B. Weissler, Existence and non-existence of global solutions for a semilinear heat equation. Israel J. Math. 38 (1981) 29–40. | Zbl | MR | DOI