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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  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 & 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.  Google Scholar

[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.  Google Scholar

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P. Baras and M. Pierre, Problèmes paraboliques semi-linéaires avec données mesures, Applicable Anal., 18 (1984), 111-149.  doi: 10.1080/00036818408839514.  Google Scholar

[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.  Google Scholar

[5]

H. Brezis and A. Friedman, Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl., 62 (1983), 73-97.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., Vol. 840, Springer-Verlag, Berlin, (1981).  Google Scholar

[10]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York-Heidelberg, (1972).  Google Scholar

[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.   Google Scholar

[12]

M. Pierre, Existence criterion of nonnegative solutions for some non monotone semilinear problems, Semesterbericht Funktionalanalysis Tübingen, Wintersemester, 1983/84,249–258. Google Scholar

[13]

F. Rothe, Global Solutions of Reaction-Diffusion Systems, Lecture Notes in Math., Vol. 1072, Springer-Verlag, Berlin, 1984. doi: 10.1007/BFb0099278.  Google Scholar

[14]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Publishing Co., Amsterdam-New York, 1978.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04631-5.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

P. Baras and M. Pierre, Problèmes paraboliques semi-linéaires avec données mesures, Applicable Anal., 18 (1984), 111-149.  doi: 10.1080/00036818408839514.  Google Scholar

[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.  Google Scholar

[5]

H. Brezis and A. Friedman, Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl., 62 (1983), 73-97.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., Vol. 840, Springer-Verlag, Berlin, (1981).  Google Scholar

[10]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York-Heidelberg, (1972).  Google Scholar

[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.   Google Scholar

[12]

M. Pierre, Existence criterion of nonnegative solutions for some non monotone semilinear problems, Semesterbericht Funktionalanalysis Tübingen, Wintersemester, 1983/84,249–258. Google Scholar

[13]

F. Rothe, Global Solutions of Reaction-Diffusion Systems, Lecture Notes in Math., Vol. 1072, Springer-Verlag, Berlin, 1984. doi: 10.1007/BFb0099278.  Google Scholar

[14]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Publishing Co., Amsterdam-New York, 1978.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04631-5.  Google Scholar

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