Optimal condition for blow-up of the critical \(L^q\) norm for the semilinear heat equation. (English) Zbl 1420.35139
Summary: We shed light on a long-standing open question for the semilinear heat equation \(u_t = \Delta u + | u |^{p - 1} u\). Namely, without any restriction on the exponent \(p > 1\) nor on the smooth domain \({\Omega}\), we prove that the critical \(L^q\) norm blows up whenever the solution undergoes type I blow-up. A similar property is also obtained for the local critical \(L^q\) norm near any blow-up point.
In view of recent results of existence of type II blow-up solutions with bounded critical \(L^q\) norm, which are counter-examples to the open question, our result seems to be essentially the best possible result in general setting. This close connection between type I blow-up and critical \(L^q\) norm blow-up appears to be a completely new observation.
Our proof is rather involved and requires the combination of various ingredients. It is based on analysis in similarity variables and suitable rescaling arguments, combined with backward uniqueness and unique continuation properties for parabolic equations.
As a by-product, we obtain the nonexistence of self-similar profiles in the critical \(L^q\) space. Such properties were up to now only known for \(p \leq p_S\) and in radially symmetric case for \(p > p_S\), where \(p_S\) is the Sobolev exponent.
In view of recent results of existence of type II blow-up solutions with bounded critical \(L^q\) norm, which are counter-examples to the open question, our result seems to be essentially the best possible result in general setting. This close connection between type I blow-up and critical \(L^q\) norm blow-up appears to be a completely new observation.
Our proof is rather involved and requires the combination of various ingredients. It is based on analysis in similarity variables and suitable rescaling arguments, combined with backward uniqueness and unique continuation properties for parabolic equations.
As a by-product, we obtain the nonexistence of self-similar profiles in the critical \(L^q\) space. Such properties were up to now only known for \(p \leq p_S\) and in radially symmetric case for \(p > p_S\), where \(p_S\) is the Sobolev exponent.
MSC:
| 35K58 | Semilinear parabolic equations |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B44 | Blow-up in context of PDEs |
| 35B33 | Critical exponents in context of PDEs |
References:
| [1] | Andreucci, D.; Herrero, M. A.; Velázquez, J. J.L., Liouville theorems and blow up behaviour in semilinear reaction diffusion systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14, 1-53 (1997) · Zbl 0877.35019 |
| [2] | H. Brezis, 1979, Unpublished manuscript.; H. Brezis, 1979, Unpublished manuscript. |
| [3] | Brezis, H.; Cazenave, Th., A nonlinear heat equation with singular initial data, J. Anal. Math., 68, 277-304 (1996) · Zbl 0868.35058 |
| [4] | Budd, C.; Qi, Y.-W., The existence of bounded solutions of a semilinear elliptic equation, J. Differential Equations, 82, 207-218 (1989) · Zbl 0709.35039 |
| [5] | Cazenave, Th.; Dickstein, F.; Naumkin, I.; Weissler, F. B., Sign-changing self-similar solutions of the nonlinear heat equation with positive initial value, Amer. J. Math. (2019), in press. Preprint |
| [6] | Collot, C., Nonradial type II blow up for the energy-supercritical semilinear heat equation, Anal. PDE, 10, 127-252 (2017) · Zbl 1372.35153 |
| [7] | Collot, C.; Merle, F.; Raphaël, P., Dynamics near the ground state for the energy critical nonlinear heat equation in large dimensions, Comm. Math. Phys., 352, 215-285 (2017) · Zbl 1401.35178 |
| [8] | Collot, C.; Merle, F.; Raphaël, P., Stability of ODE blow-up for the energy critical semilinear heat equation, C. R. Acad. Sci. Paris, Ser. I, 355, 65-79 (2017) · Zbl 1364.35156 |
| [9] | Collot, C.; Merle, F.; Raphaël, P., On strongly anisotropic type II blow up (2017), Preprint |
| [10] | Collot, C.; Raphaël, P.; Szeftel, J., On the stability of type I blow up for the energy super critical heat equation, Mem. Amer. Math. Soc., 260, 1255 (2019) · Zbl 1430.35138 |
| [11] | Del Pino, M.; Musso, M.; Wei, J., Geometry driven type II higher dimensional blow-up for the critical heat equation (2017), Preprint |
| [12] | Del Pino, M.; Musso, M.; Wei, J., Type II blow-up in the 5-dimensional energy critical heat equation, Acta Math. Sinica, 35, 1027-1042 (2019) · Zbl 1417.35074 |
| [13] | Escauriaza, L.; Fernandez, F. J., Unique continuation for parabolic operators, Ark. Mat., 41, 35-60 (2003) · Zbl 1028.35052 |
| [14] | Escauriaza, L.; Seregin, G.; Šverák, V., On backward uniqueness for parabolic equations, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 288, 100-103 (2002) · Zbl 1068.35090 |
| [15] | Escauriaza, L.; Seregin, G.; Šverák, V., On backward uniqueness for parabolic equations, Arch. Ration. Mech. Anal., 169, 147-157 (2003) · Zbl 1039.35052 |
| [16] | Escauriaza, L.; Seregin, G.; Šverák, V., Backward uniqueness for the heat operator in half space, Algebra i Analiz. Algebra i Analiz, St. Petersburg Math. J., 15, 139-148 (2004), translation in · Zbl 1053.35052 |
| [17] | Escauriaza, L.; Seregin, G.; Šverák, V., \(L_{3, \infty}\)-solutions to the Navier-Stokes equations and backward uniqueness, Uspekhi Mat. Nauk, 58, 350, 3-44 (2003) · Zbl 1064.35134 |
| [18] | Filippas, S.; Herrero, M. A.; Velázquez, J. J.L., Fast blow-up mechanisms for sign-changing solutions of a semilinear parabolic equation with critical nonlinearity, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 456, 2957-2982 (2000) · Zbl 0988.35032 |
| [19] | Friedman, A.; McLeod, B., Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34, 425-447 (1985) · Zbl 0576.35068 |
| [20] | Fujishima, Y.; Ishige, K., Blow-up set for type I blowing up solutions for a semilinear heat equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31, 231-247 (2014) · Zbl 1297.35052 |
| [21] | Giga, Y.; Kohn, R. V., Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math., 38, 297-319 (1985) · Zbl 0585.35051 |
| [22] | Giga, Y.; Kohn, R. V., Characterizing blowup using similarity variables, Indiana Univ. Math. J., 36, 1-40 (1987) · Zbl 0601.35052 |
| [23] | Giga, Y.; Kohn, R. V., Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math., 42, 845-884 (1989) · Zbl 0703.35020 |
| [24] | Giga, Y.; Matsui, S.; Sasayama, S., Blow up rate for semilinear heat equation with subcritical nonlinearity, Indiana Univ. Math. J., 53, 483-514 (2004) · Zbl 1058.35096 |
| [25] | Giga, Y.; Matsui, S.; Sasayama, S., On blow-up rate for sign-changing solutions in a convex domain, Math. Methods Appl. Sci., 27, 1771-1782 (2004) · Zbl 1066.35043 |
| [26] | Gross, L., Logarithmic Sobolev inequalities, Amer. J. Math., 97, 1061-1083 (1976) · Zbl 0318.46049 |
| [27] | Haraux, A.; Weissler, F. B., Non-uniqueness for a semilinear initial value problem, Indiana Univ. Math. J., 31, 167-189 (1982) · Zbl 0465.35049 |
| [28] | Herrero, M. A.; Velázquez, J. J.L., Blow-up behaviour of one-dimensional semilinear parabolic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10, 131-189 (1993) · Zbl 0813.35007 |
| [29] | Herrero, M. A.; Velázquez, J. J.L., Explosion de solutions d’équations paraboliques semilinéaires supercritiques, C. R. Acad. Sci. Paris, Ser. I, Math., 319, 141-145 (1994) · Zbl 0806.35005 |
| [30] | M.A. Herrero, J.J.L. Velázquez, A blow up result for semilinear heat equations in the supercritical case, Preprint, 1994.; M.A. Herrero, J.J.L. Velázquez, A blow up result for semilinear heat equations in the supercritical case, Preprint, 1994. |
| [31] | Ishige, K.; Mizoguchi, N., Blow-up behavior for semilinear heat equations with boundary conditions, Differential Integral Equations, 16, 663-690 (2003) · Zbl 1035.35052 |
| [32] | Lepin, L. A., Countable spectrum of eigenfunctions of a nonlinear heat-conduction equation with distributed parameters, Diff. Uravn.. Diff. Uravn., Differ. Equ., 24, 799-805 (1988), English translation: · Zbl 0675.35054 |
| [33] | Lepin, L. A., Self-similar solutions of a semilinear heat equation, Mat. Model., 2, 63-74 (1990), (in Russian) · Zbl 0972.35506 |
| [34] | Matano, H.; Merle, F., On nonexistence of type II blowup for a supercritical nonlinear heat equation, Comm. Pure Appl. Math., 57, 1494-1541 (2004) · Zbl 1112.35098 |
| [35] | Matano, H.; Merle, F., Classification of type I and type II behaviors for a supercritical nonlinear heat equation, J. Funct. Anal., 256, 992-1064 (2009) · Zbl 1178.35084 |
| [36] | Matos, J., Blow up of critical and subcritical norms in semilinear heat equations, Adv. Differential Equations, 3, 497-532 (1998) · Zbl 0946.35011 |
| [37] | Merle, F.; Raphaël, P.; Szeftel, J., On strongly anisotropic type I blow up, Int. Math. Res. Not. (2018) |
| [38] | Mizoguchi, N., Type-II blowup for a semilinear heat equation, Adv. Differential Equations, 9, 1279-1316 (2004) · Zbl 1122.35050 |
| [39] | Mizgouchi, N., Nonexistence of type II blowup solution for a semilinear heat equation, J. Differential Equations, 250, 26-32 (2011) · Zbl 1228.35068 |
| [40] | Mizoguchi, N., Determination of blowup type in the parabolic-parabolic Keller-Segel system, Math. Ann. (2018) |
| [41] | Poláčik, P.; Quittner, P.; Souplet, Ph., Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part II: parabolic equations, Indiana Univ. Math. J., 56, 879-908 (2007) · Zbl 1122.35051 |
| [42] | Quittner, P., Liouville theorems for scaling invariant superlinear parabolic problems with gradient structure, Math. Ann., 364, 269-292 (2016) · Zbl 1336.35193 |
| [43] | Quittner, P.; Souplet, Ph., Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts: Basel Textbooks (2007), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 1128.35003 |
| [44] | Quittner, P.; Souplet, Ph., Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts: Basel Textbooks (2019), Birkhäuser Verlag: Birkhäuser Verlag Cham · Zbl 1423.35004 |
| [45] | Schweyer, R., Type II blow-up for the four dimensional energy critical semi linear heat equation, J. Funct. Anal., 263, 3922-3983 (2012) · Zbl 1270.35137 |
| [46] | Seki, Y., Type II blow-up mechanisms in a semilinear heat equation with critical Joseph-Lundgren exponent, J. Funct. Anal., 275, 3380-3456 (2018) · Zbl 1401.35209 |
| [47] | Souplet, Ph., Single point blow-up for a semilinear parabolic system, J. Eur. Math. Soc., 11, 169-188 (2009) · Zbl 1171.35059 |
| [48] | Weissler, F. B., Local existence and nonexistence for semilinear parabolic equations in \(L^p\), Indiana Univ. Math. J., 29, 79-102 (1980) · Zbl 0443.35034 |
| [49] | Weissler, F. B., \(L^p\)-energy and blow-up for a semilinear heat equation, (Nonlinear Functional Analysis and Its Applications. Nonlinear Functional Analysis and Its Applications, Proc. Sympos. Pure Math., vol. 45, Part 2 (1986), American Mathematical Society: American Mathematical Society Providence, RI), 545-551 · Zbl 0631.35049 |
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