Initial trace of solutions of semilinear heat equations with absorption. (English) Zbl 1282.35194
Summary: We study the initial trace problem for positive solutions of semilinear heat equations with strong absorption. We show that in general this initial trace is an outer regular Borel measure. We emphasize in particular the case where \(u\) satisfies \(\partial _tu-\varDelta u+t^\alpha| u|^{q-1}u=0\), with \(q>1\) and \(\alpha >-1\) and prove that in the subcritical case \(1<q<q_{\alpha, N}:= 1+2(1+\alpha )/N\) the initial trace establishes a one to one correspondence between the set of outer regular Borel measures in \(\mathbb R^N\) and the set of positive solutions in \(\mathbb R^N\times\mathbb R_+\).
MSC:
| 35K58 | Semilinear parabolic equations |
| 35K15 | Initial value problems for second-order parabolic equations |
References:
| [1] | Marcus, M.; Véron, L., The boundary trace and generalized boundary value problem for semilinear elliptic equations with coercive absorption, Comm. Pure Appl. Math., LVI, 689-731 (2003) · Zbl 1121.35314 |
| [2] | Brezis, H.; Friedman, A., Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl. (9), 62, 73-97 (1983) · Zbl 0527.35043 |
| [3] | Brezis, H.; Peletier, L. A.; Terman, D., A very singular solution of the heat equation with absorption, Arch. Ration. Mech. Anal., 95, 185-209 (1986) · Zbl 0627.35046 |
| [4] | Kamin, S.; Peletier, L. A., Large time behaviour of solutions of the heat equation with absorption, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 12, 393-408 (1985) · Zbl 0598.35050 |
| [5] | Marcus, M.; Véron, L., Initial trace of positive solutions of some nonlinear parabolic equations, Comm. Partial Differential Equations, 24, 1445-1499 (1999) · Zbl 1059.35054 |
| [6] | Marcus, M.; Véron, L., Initial trace of positive solutions to semilinear parabolic inequalities, Adv. Nonlinear Stud., 2, 395-436 (2002) · Zbl 1021.35051 |
| [7] | Oswald, L., Isolated positive singularities for a nonlinear heat equation, Houston J. Math., 14, 543-572 (1988) · Zbl 0696.35023 |
| [8] | Moutoussamy, I.; Véron, L., Isolated singularities and asymptotic behaviour of the solutions of a semi-linear heat equation, Asymptot. Anal., 9, 259-289 (1994) · Zbl 0812.35058 |
| [9] | Marcus, M.; Véron, L., Semilinear parabolic equations with measure boundary data and isolated singularities, J. Anal. Math., 85, 245-290 (2001) · Zbl 1001.35065 |
| [10] | Véron, L., Semilinear elliptic equations with uniform blow-up on the boundary, J. Anal. Math., 59, 231-250 (1992) · Zbl 0802.35042 |
| [11] | Shishkov, A.; Véron, L., The balance between diffusion and absorption in semilinear parabolic equations, Rend. Lincei Mat., 18, 59-96 (2007) · Zbl 1139.35366 |
| [12] | Escobedo, M.; Kavian, O., Variational problems related to self-similar solutions of the heat equation, Nonlinear Anal., 11, 1103-1133 (1987) · Zbl 0639.35038 |
| [13] | Gkikas, K.; Véron, L., Initial value problems for diffusion equations with singular potential, Contemp. Math., 594, 201-230 (2013) · Zbl 1322.35089 |
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