Journal of Elliptic and Parabolic Equations

, Volume 4, Issue 1, pp 141–176

# The nonlinear heat equation involving highly singular initial values and new blowup and life span results

Article

## Abstract

In this paper we prove local existence of solutions to the nonlinear heat equation $$u_t = \Delta u +a |u|^\alpha u, \; t\in (0,T),\; x=(x_1,\ldots , x_N)\in {\mathbb {R}}^N,\; a = \pm 1,\; \alpha >0;$$ with initial value $$u(0)\in L^1_{\mathrm{{loc}}}({\mathbb {R}}^N{\setminus }\{0\}),$$ anti-symmetric with respect to $$x_1,\; x_2,\ldots , x_m$$ and $$|u(0)|\le C(-1)^m\partial _{1}\partial _{2}\cdots \partial _{m}(|x|^{-\gamma })$$ for $$x_1>0,\ldots , x_m>0,$$ where $$C>0$$ is a constant, $$m\in \{1, 2,\ldots , N\},$$ $$0<\gamma <N$$ and $$0<\alpha <2/(\gamma +m).$$ This gives a local existence result with highly singular initial values. As an application, for $$a=1,$$ we establish new blowup criteria for $$0<\alpha \le 2/(\gamma +m),$$ including the case $$m=0.$$ Moreover, if $$(N-4)\alpha <2,$$ we prove the existence of initial values $$u_0 = \lambda f,$$ for which the resulting solution blows up in finite time $$T_{\max }(\lambda f),$$ if $$\lambda >0$$ is sufficiently small. We also construct blowing up solutions with initial data $$\lambda _n f$$ such that $$\lambda _n^{[({1\over \alpha }-{\gamma +m\over 2})^{-1}]}T_{\max }(\lambda _n f)$$ has different finite limits along different sequences $$\lambda _n\rightarrow 0.$$ Our result extends the known “small lambda” blow up results for new values of $$\alpha$$ and a new class of initial data.

## Keywords

Nonlinear heat equation Highly singular initial values Finite time blow-up

## Mathematics Subject Classification

Primary 35K55 35A01 35B44 Secondary 35K57 35C15

## References

1. 1.
Bandle, C., Levine, H.A.: On the existence and nonexistence of global solutions of reaction–diffusion equations in sectorial domains. Trans. AMS 316, 595–622 (1989)
2. 2.
Brézis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York (2011)Google Scholar
3. 3.
Brézis, H., Friedman, A.: Nonlinear parabolic equations involving measures as initial conditions. J. Math. Pures Appl. 62, 73–97 (1983)
4. 4.
Cazenave, T., Dickstein, F., Escobedo, M., Weissler, F.B.: Self-similar solutions of a nonlinear heat equation. J. Math. Sci. Univ. Tokyo 8, 501–540 (2001)
5. 5.
Cazenave, T., Dickstein, F., Weissler, F.B.: Universal solutions of the heat equation in $${\mathbb{R}}^{N}$$. Discrete Contin. Dyn. Syst. 9, 1105–1132 (2003)
6. 6.
Cazenave, T., Dickstein, F., Weissler, F.B.: Multi-scale multi-profile global solutions of parabolic equations in $${\mathbb{R}}^{N}$$. Discrete Contin. Dyn. Syst. Ser. S 5, 449–472 (2012)
7. 7.
Dickstein, F.: Blowup stability of solutions of the nonlinear heat equation with a large life span. J. Differ. Equ. 223, 303–328 (2006)
8. 8.
Fermanian Kammerer, C., Merle, F., Zaag, H.: Stability of the blow-up profile of nonlinear heat equations from the dynamical system point of view. Math. Ann. 317, 347–387 (2000)
9. 9.
Ghoul, T.: An extension of Dickstein’s “small lambda” theorem for finite time blowup. Nonlinear Anal. T.M.A. 74, 6105–6115 (2011)
10. 10.
Giga, Y., Matsui, S., Sasayama, S.: Blow up rate for semilinear heat equations with subcritical nonlinearity. Indiana Univ. Math. J. 53, 483–514 (2004)
11. 11.
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)
12. 12.
Gui, C., Wang, X.: Life span of solutions of the Cauchy problem for a semilinear heat equation. J. Differ. Equ. 115, 166–172 (1995)
13. 13.
Kavian, O.: Remarks on the large time behaviour of a nonlinear diffusion equation. Ann. I. H. Poincaré Anal. Non Linéaire 4, 423–452 (1987)
14. 14.
Lee, T.Y., Ni, W.M.: Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem. Trans. Am. Math. Soc. 333, 365–378 (1992)
15. 15.
Levine, H.A., Meier, P.: The value of the critical exponent for reaction–diffusion equations in cones. Arch. Ration. Mech. Anal. 109, 73–80 (1990)
16. 16.
Meier, P.: Existence et non-existence de solutions globales d’une équations de la chaleur semi-linéaire: extention d’un théoreme de Fujita. C. R. Acad. Sci. Paris Ser. I(303), 635–637 (1986)
17. 17.
Meier, P.: Blow up of solutions of semilinear parabolic differential equations. Z. Angew. Math. Phys. 39, 135–149 (1988)
18. 18.
Mizoguchi, N., Yanagida, E.: Blowup and life span of solutions for a semilinear parabolic equation. SIAM J. Math. Anal. 29, 1434–1446 (1998)
19. 19.
Molinet, L., Tayachi, S.: Remarks on the Cauchy problem for the one-dimensional quadratic (fractional) heat equation. J. Funct. Anal. 269, 2305–2327 (2015)
20. 20.
Mouajria, H., Tayachi, S., Weissler, F.B.: The heat equation on sectorial domains, highly singular initial values and applications. J. Evol. Equ. 16, 341–364 (2016)
21. 21.
Mueller, C.E., Weissler, F.B.: Single point blow-up for a general semilinear heat equation. Indiana Univ. Math. J. 34, 881–913 (1985)
22. 22.
Quittner, P., Souplet, Ph: Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States. Birkhäuser, Basel (2007)
23. 23.
Ribaud, F.: Semilinear parabolic equations with distributions as initial values. Discrete Contin. Dyn. Syst. 3, 305–316 (1997)
24. 24.
Rouchon, P.: Blow-up of solutions of nonlinear heat equations in unbounded domains for slowly decaying initial data. ZAMP 52, 1017–1032 (2001)
25. 25.
Souplet, Ph, Weissler, F.B.: Self-similar sub-solutions and blow-up for nonlinear parabolic equations. J. Math. Anal. Appl. 212, 60–74 (1997)
26. 26.
Tayachi, S., Weissler, F.B.: The nonlinear heat equation with high order mixed derivatives of the Dirac delta as initial values. Trans. AMS 366, 505–530 (2014)
27. 27.
Tayachi, S., Weissler, F.B.: Some remarks on life span results (in preparation) Google Scholar
28. 28.
Weissler, F.B.: Existence and nonexistence of global solutions for a semilinear heat equation. Isr. J. Math. 38, 29–40 (1981)
29. 29.
Weissler, F.B.: $$L^p$$-energy and blow-up for a semilinear heat equation. Proc. Symp. Pure Math. 45(Part 2), 545–551 (1986)
30. 30.
Wu, J.: Well-posedness of a semilinear heat equation with weak initial data. J. Fourier Anal. Appl. 4, 629–642 (1998)