Blowup versus global in time existence of solutions for nonlinear heat equations

Piotr Cezary Biler



This note is devoted to a simple proof of blowup of solutions for a nonlinear heat equation. The criterion for a blowup is expressed in terms of a Morrey space norm and is in a sense complementary to conditions guaranteeing the global in time existence of solutions. The method goes back to H. Fujita and extends to other nonlinear parabolic equations.


Nonlinear heat equation; blowup of solutions; global existence of solutions

Full Text:



D. Andreucci and E. DiBenedetto, On the Cauchy problem and initial traces for a class of evolution equations with strongly nonlinear sources, Ann. Sci. Norm. Super. Pisa Cl. Sci. (4) 18 (1991), 363–441.

P. Baras and and M. Pierre, Critère d’existence de solutions positives pour des équations semi-linéaires non monotones, Ann. Inst. H. Poincaré Anal. non Linéaire 2 (1985), 185–212.

P. Biler, The Cauchy problem and self-similar solutions for a nonlinear parabolic equation, Studia Math. 114 (1995), 181–205.

P. Biler, Blowup of solutions for nonlinear nonlocal heat equations, arXiv:1807.03569, 1–12. (submitted)

P. Biler, T. Cieślak, G. Karch and J. Zienkiewicz, Local criteria for blowup of solutions in two-dimensional chemotaxis models, Discrete Contin. Dyn. Syst. A 37 (2017), 1841–1856.

P. Biler and G. Karch, Blowup of solutions to generalized Keller–Segel model, J. Evol. Equ. 10 (2010), 247–262.

P. Biler, G. Karch and D. Pilarczyk, Global radial solutions in classical Keller–Segel chemotaxis model, arXiv:1807.02628. 1–20. (submitted)

P. Biler, G. Karch and J. Zienkiewicz, Optimal criteria for blowup of radial and N -symmetric solutions of chemotaxis systems, Nonlinearity 28 (2015), 4369–4387.

P. Biler, G. Karch and J. Zienkiewicz, Morrey spaces norms and criteria for blowup in chemotaxis models, Networks and Non Homogeneous Media 11 (2016), 239–250.

P. Biler, G. Karch and J. Zienkiewicz, Large global-in-time solutions to a nonlocal model of chemotaxis, Adv. Math. 330 (2018), 834–875.

P. Biler and D. Pilarczyk, Around a singular solution to a nonlocal nonlinear heat equation, arXiv:1807.03567, 1–20. (submitted)

P. Biler and J. Zienkiewicz, Blowing up radial solutions in the minimal Keller–Segel chemotaxis model, arXiv:1807.02633, 1–20. (submitted).

M.P. Brenner, P. Constantin, L.P. Kadanoff, A. Schenkel and S.C. Venkataramani, Diffusion, attraction and collapse, Nonlinearity 12 (1999), 1071–1098.

J. Burczak and R. Granero-Belinchón, Global solutions for a supercritical driftdiffusion equation, Adv. Math. 295 (2016), 334–367.

P. Constantin, F. Gancedo, R. Shvydkoy and V. Vicol, Global regularity for 2D Muskat equations with finite slope, Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), 1041–1074.

A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Commun. Math. Phys. 249 (2004), 511–528.

P. Constantin and V. Vicol, Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geom. Funct. Anal. 22 (2012), 1289–1321.

H. Fujita, On the blowing up of solutions of the Cauchy problem for ut = ∆u + u1+α , J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966), 109–124.

V.A. Galaktionov and J.L. Vázquez, Continuation of blowup solutions of nonlinear heat equations in several space dimensions, Comm. Pure Appl. Math. 50 (1997), 1–67.

Y. Giga and R.V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math. 38 (1985), 297–319.

Y. Giga and R.V. Kohn, Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math. 42 (1989), 845–884.

R. Granero-Belinchón and R. Orive-Illera, An aggregation equation with a nonlocal flux, Nonlinear Anal. 108 (2014), 260–274.

C. Gui, W.-M. Ni and X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in Rn , Comm. Pure Appl. Math. 45 (1992), 1153–1181.

T.-Y. Lee and W.-M. Ni, Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem, Trans. Amer. Math. Soc. 333 (1992), 365–378.

P.-G. Lemarié-Rieusset, Small data in an optimal Banach space for the parabolic–parabolic and parabolic–elliptic Keller–Segel equations in the whole space, Adv. Differential Equations 18 (2013), 1189–1208.

X. Li and Z. Xiang, Existence and nonexistence of local/global solutions for a nonhomogeneous heat equation, Comm. Pure Appl. Analysis 13 (2014), 1465–1480.

N. Mizoguchi, On the behavior of solutions for a semilinear parabolic equation with supercritical nonlinearity, Math. Z. 239 (2002), 215–229.

N. Mizoguchi, Blowup behavior of solutions for a semilinear heat equation with supercritical nonlinearity, J. Differential Equations 205 (2004), 298–328.

N. Mizoguchi, Boundedness of global solutions for a supercritical semilinear heat equation and its application, Indiana Univ. Math. J. 54 (2005), 1047–1059.

P. Poláčik and E. Yanagida, On bounded and unbounded global solutions of a supercritical semilinear heat equation, Math. Ann. 327 (2003), 745–771.

P. Poláčik and E. Yanagida, Global unbounded solutions of the Fujita equation in the intermediate range, Math. Ann. 360 (2014), 255–266.

P. Quittner and Ph. Souplet, Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States, Birkhäuser Advanced Texts, 2007.

Ph. Souplet, Morrey spaces and classification of global solutions for a supercritical semilinear heat equation in Rn , J. Funct. Anal. 272 (2017), 2005–2037.

Ph. Souplet and F.B. Weissler, Regular self-similar solutions of the nonlinear heat equation with initial data above the singular steady state, Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003), 213–235.

M.E. Taylor, Analysis on Morrey spaces and applications to Navier–Stokes and other evolution equations, Comm. Partial Differential Equations 17 (1992), 1407–1456.

F.G. Tricomi and A. Erdélyi, The asymptotic expansion of a ratio of Gamma functions, Pacific J. Math. 1 (1951), 133–142.

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

F.B. Weissler, Lp -energy and blow-up for a semilinear heat equation, Proc. Symp. Pure Math. 45 (1986), no. 2, 545–551.


  • There are currently no refbacks.

Partnerzy platformy czasopism