On Nonexistence of Baras--Goldstein Type without Positivity Assumptions for Singular Linear and Nonlinear Parabolic Equations

V. A. Galaktionov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function theory and nonlinear partial differential equations, Tome 260 (2008), pp. 130-150.

Résumé

The celebrated result by Baras and Goldstein (1984) established that the heat equation with the inverse square potential in the unit ball $B_1\subset\mathbb R^N$, $N\ge3$, $u_t=\Delta u+\frac c{|x|^2}u$ in $B_1\times(0,T)$, $u|_{\partial B_1}=0$, in the supercritical range $c>c_\mathrm{Hardy}=\bigl(\frac{N-2}2\bigr)^2$ does not have a solution for any nontrivial $L^1$ initial data $u_0(x)\ge0$ in $B_1$ (or for a positive measure $u_0$). More precisely, it was proved that a regular approximation of a possible solution by a sequence $\{u_n(x,t)\}$ of classical solutions corresponding to truncated bounded potentials given by $V(x)=\frac c{|x|^2}\mapsto V_n(x)=\min\bigl \{\frac c{|x|^2},n\bigr\}$ ($n\ge1$) diverges; i.e., as $n\to\infty$, $u_n(x,t)\to+\infty$ in $B_1\times(0,T)$. Similar features of “nonexistence via approximation” for semilinear heat PDEs were inherent in related results by Brezis–Friedman (1983) and Baras–Cohen (1987). The main goal of this paper is to justify that this nonexistence result has wider nature and remains true without the positivity assumption on data $u_0(x)$ that are assumed to be regular and positive at $x=0$. Moreover, nonexistence as the impossibility of regular approximations of solutions is true for a wide class of singular nonlinear parabolic problems as well as for higher order PDEs including, e.g., $u_t =\Delta(|u|^{m-1}u)+\frac{|u|^{p-1}u}{|x|^2}$, $m\ge1$, $p>1$, and $u_t=-\Delta^2u+\frac c{|x|^4}u$, $c>c_\mathrm H=\bigl[\frac{N(N-4)}4\bigr]^2$, $N>4$.

Bibliographie

[1] Angenent S., “The zero set of a solution of a parabolic equation”, J. reine und angew. Math., 390 (1988), 79–96 | MR | Zbl

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

[3] Baras P., Cohen L., “Sur l'explosion totale après $T_{\mathrm{max}}$ de la solution d'une équation de la chaleur semi-linéaire”, C. r. Acad. sci. Paris. Sér. 1, 300 (1985), 295–298 | MR | Zbl

[4] Baras P., Goldstein J. A., “The heat equation with a singular potential”, Trans. Amer. Math. Soc., 284 (1984), 121–139 | DOI | MR | Zbl

[5] Birman M. Sh., Solomyak M. Z., Spektralnaya teoriya samosopryazhennykh operatorov v gilbertovom prostranstve, Izd-vo LGU, L., 1980 | MR

[6] Brezis H., Cabré X., “Some simple nonlinear PDE's without solutions”, Boll. Unione mat. Ital. B: Artic. ric. mat., 1:2 (1998), 223–262 | MR | Zbl

[7] Brezis H., Cazenave T., Martel Y., Ramiandrisoa A., “Blow up for $u_t-\Delta u=g(u)$ revisited”, Adv. Diff. Equat., 1 (1996), 73–90 | MR | Zbl

[8] Brezis H., Friedman A., “Nonlinear parabolic equations involving measures as initial conditions”, J. math. pures et appl., 62 (1983), 73–97 | MR | Zbl

[9] Coddington E. A., Levinson N., Theory of ordinary differential equations, McGraw-Hill, New York, London, 1955 | MR | Zbl

[10] Dold J. W., Galaktionov V. A., Lacey A. A., Vázquez J. L., “Rate of approach to a singular steady state in quasilinear reaction-diffusion equations”, Ann. Scuola Norm. Super. Pisa. Cl. Sci. Ser. 4, 26 (1998), 663–687 | MR | Zbl

[11] Fila M., Matano H., Poláčik P., “Immediate regularization after blow-up”, SIAM J. Math. Anal., 37 (2005), 752–776 | DOI | MR | Zbl

[12] Galaktionov V. A., Geometric Sturmian theory of nonlinear parabolic equations and applications, Chapman Hall/CRC, Boca Raton, FL, 2004 | MR | Zbl

[13] Galaktionov V. A., “On extensions of Hardy's inequalities”, Commun. Contemp. Math., 7 (2005), 97–120 | DOI | MR | Zbl

[14] Galaktionov V. A., “On extensions of higher-order Hardy's inequalities”, Diff. and Integr. Equat., 19 (2006), 327–344 | MR

[15] Galaktionov V. A., Kamotski I. V., “On nonexistence of Baras–Goldstein type for higher-order parabolic equations with singular potentials”, Trans. Amer. Math. Soc. (to appear)

[16] Galaktionov V. A., Shishkov A. E., “Higher-order quasilinear parabolic equations with singular initial data”, Commun. Contemp. Math., 8 (2006), 331–354 | DOI | MR | Zbl

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

[18] Goldstein G. R., Goldstein J. A., Kombe I., “Nonlinear parabolic equations with singular coefficient and critical exponent”, Appl. Anal., 84 (2005), 571–583 | DOI | MR | Zbl

[19] Goldstein J. A., Zhang Q. S., “On a degenerate heat equation with a singular potential”, J. Funct. Anal., 186 (2001), 342–359 | DOI | MR | Zbl

[20] Goldstein J. A., Zhang Q. S., “Linear parabolic equations with strong singular potentials”, Trans. Amer. Math. Soc., 355 (2003), 197–211 | DOI | MR | Zbl

[21] Hale J. K., Asymptotic behavior of dissipative systems, Amer. Math. Soc., Providence, RI, 1988 | MR | Zbl

[22] Hulshof J., “Similarity solutions of the porous medium equation with sign changes”, J. Math. Anal. and Appl., 157 (1991), 75–111 | DOI | MR | Zbl

[23] Kato T., Perturbation theory for linear operators, Springer–Verlag, Berlin, New York, 1976 | MR

[24] Kombe I., “The linear heat equation with highly oscillating potential”, Proc. Amer. Math. Soc., 132 (2004), 2683–2691 | DOI | MR | Zbl

[25] Kombe I., “Nonlinear degenerate parabolic equations for Baouendi–Grushin operators”, Math. Nachr., 279 (2006), 756–773 | DOI | MR | Zbl

[26] Martel Y., “Complete blow up and global behaviour of solutions of $u_t-\Delta u=g(u)$”, Ann. Inst. H. Poincaré. Anal. non lin., 15 (1998), 687–723 | DOI | MR | Zbl

[27] Mitidieri E., Pokhozhaev S. I., Apriornye otsenki i otsutstvie reshenii nelineinykh uravnenii i neravenstv v chastnykh proizvodnykh, Tr. MIAN, 234, Nauka, M., 2001 | MR

[28] Naimark M. A., Lineinye differentsialnye operatory, Gostekhizdat, M., 1954

[29] Pisante A., “Hardy inequalities and dynamic instability of singular Yamabe metrics”, Ann. Inst. H. Poincaré. Anal. non lin., 23 (2006), 591–628 | DOI | MR | Zbl

[30] Sturm C., “Mémoire sur une classe d'équations à différences partielles”, J. math. pures et appl., 1 (1836), 373–444