A semilinear heat equation with initial data in negative Sobolev spaces

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February 2021, 14(2): 745-767. doi: 10.3934/dcdss.2020365

A semilinear heat equation with initial data in negative Sobolev spaces

Haruki Umakoshi

Kusauchi Yamashina 53-7, Kyotanabe City, Kyoto, Japan

Received December 2019 Revised January 2020 Published February 2021 Early access May 2020

We give a sufficient conditions for the existence, locally in time, of solutions to semilinear heat equations with nonlinearities of type $ |u|^{p-1}u $, when the initial datas are in negative Sobolev spaces $ H_q^{-s}(\Omega) $, $ \Omega \subset \mathbb{R}^N $, $ s \in [0,2] $, $ q \in (1,\infty) $. Existence is for instance proved for $ q>\frac{N}{2}\left(\frac{1}{p-1}-\frac{s}{2}\right)^{-1} $. This is an extension to $ s \in (0,2] $ of previous results known for $ s = 0 $ with the critical value $ \frac{N(p-1)}{2} $. We also observe the uniqueness of solutions in some appropriate class.

Keywords: Semilinear heat equation, singular initial data.

Mathematics Subject Classification: Primary: 35K91, 49K40; Secondary: 35B65.

Citation: Haruki Umakoshi. A semilinear heat equation with initial data in negative Sobolev spaces. Discrete and Continuous Dynamical Systems - S, 2021, 14 (2) : 745-767. doi: 10.3934/dcdss.2020365

References:

[1]	P. Baras, Non-unicité des solutions d'une équation d'évolution non-linéaire, Ann. Fac. Sci. Toulouse Math., 5 (1983), 287-302. doi: 10.5802/afst.600.
[2]	P. Baras 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. doi: 10.1016/S0294-1449(16)30402-4.
[3]	P. Baras and M. Pierre, Problèmes paraboliques semi-linéaires avec données mesures, Applicable Anal., 18 (1984), 111-149. doi: 10.1080/00036818408839514.
[4]	H. Brezis and T. Cazenave, A nonlinear heat equation with singular initial data, J. Anal. Math., 68 (1996), 277-304. doi: 10.1007/BF02790212.
[5]	H. Brezis and A. Friedman, Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl., 62 (1983), 73-97.
[6]	M. Cowling, I. Doust, A. Mcintosh and A. Yagi, Banach space operators with a bounded $H^\infty$ functional calculus, J. Austral. Math. Soc. Ser. A, 60 (1996), 59-89. doi: 10.1017/S1446788700037393.
[7]	X. T. Duong, $H^\infty$ functional calculus of second order elliptic partial differntial operators on $L^p$ spaces, Miniconference on Operators in Analysis (Sydney, 1989), Proc. Centre Math. Anal. Austral. Nat. Univ., Vol. 24, Austral. Nat. Univ., Canberra, 1990, 91–102.
[8]	A. Haraux and F. B. Weissler, Nonuniqueness for a semilinear initial value problem, Indiana Univ. Math. J., 31 (1982), 167-189. doi: 10.1512/iumj.1982.31.31016.
[9]	D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., Vol. 840, Springer-Verlag, Berlin, (1981).
[10]	J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York-Heidelberg, (1972).
[11]	E. Nakaguchi and K. Osaki, Global existence of solutions to an $n$-dimensional parabolic-parabolic system for chemotaxis with logistic-type growth and superlinear production, Osaka J. Math., 55 (2018), 51-70.
[12]	M. Pierre, Existence criterion of nonnegative solutions for some non monotone semilinear problems, Semesterbericht Funktionalanalysis Tübingen, Wintersemester, 1983/84,249–258.
[13]	F. Rothe, Global Solutions of Reaction-Diffusion Systems, Lecture Notes in Math., Vol. 1072, Springer-Verlag, Berlin, 1984. doi: 10.1007/BFb0099278.
[14]	H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Publishing Co., Amsterdam-New York, 1978.
[15]	F. B. Weissler, Local existence and nonexistence for semilinear parabolic equations in $L^p$, Indiana Univ. Math. J., 29 (1980), 79-102. doi: 10.1512/iumj.1980.29.29007.
[16]	A. Yagi, $H^\infty$ Functional Calculus and Characterization of Domains of Fractional Powers, in Oper. Theory Adv. Appl., Vol. 187, 2008,217–235. doi: 10.1007/978-3-7643-8893-5_15.
[17]	A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04631-5.

show all references

References:

[1]	P. Baras, Non-unicité des solutions d'une équation d'évolution non-linéaire, Ann. Fac. Sci. Toulouse Math., 5 (1983), 287-302. doi: 10.5802/afst.600.
[2]	P. Baras 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. doi: 10.1016/S0294-1449(16)30402-4.
[3]	P. Baras and M. Pierre, Problèmes paraboliques semi-linéaires avec données mesures, Applicable Anal., 18 (1984), 111-149. doi: 10.1080/00036818408839514.
[4]	H. Brezis and T. Cazenave, A nonlinear heat equation with singular initial data, J. Anal. Math., 68 (1996), 277-304. doi: 10.1007/BF02790212.
[5]	H. Brezis and A. Friedman, Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl., 62 (1983), 73-97.
[6]	M. Cowling, I. Doust, A. Mcintosh and A. Yagi, Banach space operators with a bounded $H^\infty$ functional calculus, J. Austral. Math. Soc. Ser. A, 60 (1996), 59-89. doi: 10.1017/S1446788700037393.
[7]	X. T. Duong, $H^\infty$ functional calculus of second order elliptic partial differntial operators on $L^p$ spaces, Miniconference on Operators in Analysis (Sydney, 1989), Proc. Centre Math. Anal. Austral. Nat. Univ., Vol. 24, Austral. Nat. Univ., Canberra, 1990, 91–102.
[8]	A. Haraux and F. B. Weissler, Nonuniqueness for a semilinear initial value problem, Indiana Univ. Math. J., 31 (1982), 167-189. doi: 10.1512/iumj.1982.31.31016.
[9]	D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., Vol. 840, Springer-Verlag, Berlin, (1981).
[10]	J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York-Heidelberg, (1972).
[11]	E. Nakaguchi and K. Osaki, Global existence of solutions to an $n$-dimensional parabolic-parabolic system for chemotaxis with logistic-type growth and superlinear production, Osaka J. Math., 55 (2018), 51-70.
[12]	M. Pierre, Existence criterion of nonnegative solutions for some non monotone semilinear problems, Semesterbericht Funktionalanalysis Tübingen, Wintersemester, 1983/84,249–258.
[13]	F. Rothe, Global Solutions of Reaction-Diffusion Systems, Lecture Notes in Math., Vol. 1072, Springer-Verlag, Berlin, 1984. doi: 10.1007/BFb0099278.
[14]	H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Publishing Co., Amsterdam-New York, 1978.
[15]	F. B. Weissler, Local existence and nonexistence for semilinear parabolic equations in $L^p$, Indiana Univ. Math. J., 29 (1980), 79-102. doi: 10.1512/iumj.1980.29.29007.
[16]	A. Yagi, $H^\infty$ Functional Calculus and Characterization of Domains of Fractional Powers, in Oper. Theory Adv. Appl., Vol. 187, 2008,217–235. doi: 10.1007/978-3-7643-8893-5_15.
[17]	A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04631-5.

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