Well-posedness of semilinear heat equations in L ¹

Laister, R.; Sierżęga, M.

Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 3, pp. 709-725.

The problem of obtaining necessary and sufficient conditions for local existence of non-negative solutions in Lebesgue spaces for semilinear heat equations having monotonically increasing source term f has only recently been resolved (Laister et al. (2016)). There, for the more difficult case of initial data in $L^1$, a necessary and sufficient integral condition on f emerged. Here, subject to this integral condition, we consider other fundamental properties of solutions with $L^1$ initial data of indefinite sign, namely: uniqueness, regularity, continuous dependence and comparison. We also establish sufficient conditions for the global-in-time continuation of solutions for small initial data in $L^1$.

[1] Arrieta, J.; Carvalho, A. Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations, Trans. Am. Math. Soc., Volume 352 (2000), pp. 285–310 | Zbl

[2] Baras, P.; Pierre, M. Critère d'existence de solutions positives pour des équations semi-linéaires non monotones, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 2 (1985), pp. 185–212 | DOI | Numdam | MR | Zbl

[3] Brézis, H.; Cazenave, T. A nonlinear heat equation with singular initial data, J. Anal. Math., Volume 68 (1996), pp. 277–304 | DOI | Zbl

[4] Celik, C.; Zhou, Z. No local $L^1$ solution for a nonlinear heat equation, Commun. Partial Differ. Equ., Volume 28 (2003), pp. 1807–1831 | DOI | Zbl

[5] Fila, M.; King, J.R.; Winkler, M.; Yanagida, E. Grow-up rate of solutions of a semilinear parabolic equation with a critical exponent, Adv. Differ. Equ., Volume 12 (2007), pp. 1–26 | Zbl

[6] 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., Volume 456 (2000), pp. 2957–2982 | Zbl

[7] Fujishima, Y.; Ioku, N. Existence and nonexistence of solutions for the heat equation with a superlinear source term, J. Math. Pures Appl., Volume 118 (2018), pp. 128–158 | DOI | Zbl

[8] Fujita, H. On the blowing up of solutions of the Cauchy problem for $u_t=\mathrm{\Delta }u+u^{1+\alpha }$ , J. Sci. Univ. Tokyo, Sect. I, Volume 13 (1966), pp. 109–124 | Zbl

[9] Hayakawa, K. On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Jpn. Acad., Ser. A, Math. Sci., Volume 49 (1973), pp. 503–505 | DOI | Zbl

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

[11] Kobayashi, K.; Sirao, T.; Tanaka, H. On the growing up problem for semilinear heat equations, J. Math. Soc. Jpn., Volume 29 (1977) no. 3, pp. 407–425 | DOI | Zbl

[12] Laister, R.; Robinson, J.C.; Sierżęga, M.; Vidal-López, A. A complete characterisation of local existence for semilinear parabolic equations in Lebesgue spaces, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 33 (2016) no. 6, pp. 1519–1538 | Numdam | Zbl

[13] Li, K. A characteristic of local existence for nonlinear fractional heat equations in Lebesgue spaces, Comput. Math. Appl., Volume 73 (2017), pp. 653–665 | Zbl

[14] Matos, J.; Terraneo, E. Nonuniqueness for a critical nonlinear heat equation with any initial data, Nonlinear Anal., Volume 55 (2003), pp. 927–936 | DOI | Zbl

[15] Mitrinović, D.S.; Pečarić, J.; Fink, A.M. Inequalities Involving Functions and Their Integrals and Derivatives, Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, 1991 | DOI | Zbl

[16] Ni, W-M.; Sacks, P. Singular behavior in nonlinear parabolic equations, Trans. Am. Math. Soc., Volume 287 (1985), pp. 657–671 | Zbl

[17] Poláčik, P.; Yanagida, E. On bounded and unbounded global solutions of a supercritical semilinear heat equation, Math. Ann., Volume 327 (2003), pp. 745–771 | Zbl

[18] Quittner, P.; Souplet, P. Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts, Birkhäuser, Basel, 2019 | Zbl

[19] Robinson, J.C.; Sierżęga, M. Supersolutions for a class of semilinear heat equations, Rev. Mat. Complut., Volume 26 (2013), pp. 341–360 | DOI | Zbl

[20] Sugitani, S. On nonexistence of global solutions for some nonlinear integral equations, Osaka J. Math., Volume 12 (1975), pp. 45–51 | Zbl

[21] Weissler, F.B. Semilinear evolution equations in Banach spaces, J. Funct. Anal., Volume 32 (1979), pp. 277–296 | DOI | Zbl

[22] Weissler, F.B. Local existence and nonexistence for semilinear parabolic equations in $L^p$ , Indiana Univ. Math. J., Volume 29 (1980), pp. 79–102 | DOI | Zbl

[23] Weissler, F.B. Existence and nonexistence of global solutions for a semilinear heat equation, Isr. J. Math., Volume 38 (1981), pp. 29–40 | DOI | Zbl