Umakoshi, Haruki A semilinear heat equation with initial data in negative Sobolev spaces. (English) Zbl 1458.35229 Discrete Contin. Dyn. Syst., Ser. S 14, No. 2, 745-767 (2021). Summary: 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. Cited in 3 Documents MSC: 35K58 Semilinear parabolic equations 35K91 Semilinear parabolic equations with Laplacian, bi-Laplacian or poly-Laplacian 35K20 Initial-boundary value problems for second-order parabolic equations 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness 35B65 Smoothness and regularity of solutions to PDEs 35D30 Weak solutions to PDEs Keywords:semilinear heat equation; singular initial data PDFBibTeX XMLCite \textit{H. Umakoshi}, Discrete Contin. Dyn. Syst., Ser. S 14, No. 2, 745--767 (2021; Zbl 1458.35229) Full Text: DOI References: [1] P. Baras, Non-unicité des solutions d’une équation d’évolution non-linéaire, Ann. Fac. Sci. Toulouse Math., 5, 287-302 (1983) · Zbl 0553.35046 [2] P. Baras; M. Pierre, Critère d’existence de solutions positives pour des équations semi-linéaires non monotones, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2, 185-212 (1985) · Zbl 0599.35073 [3] P. Baras; M. Pierre, Problèmes paraboliques semi-linéaires avec données mesures, Applicable Anal., 18, 111-149 (1984) · Zbl 0582.35060 [4] H. Brezis; T. Cazenave, A nonlinear heat equation with singular initial data, J. Anal. Math., 68, 277-304 (1996) · Zbl 0868.35058 [5] H. Brezis; A. Friedman, Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl., 62, 73-97 (1983) · Zbl 0527.35043 [6] M. Cowling; I. Doust; A. Mcintosh; A. Yagi, Banach space operators with a bounded \(H^\infty\) functional calculus, J. Austral. Math. Soc. Ser. A, 60, 59-89 (1996) · Zbl 0853.47010 [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; F. B. Weissler, Nonuniqueness for a semilinear initial value problem, Indiana Univ. Math. J., 31, 167-189 (1982) · Zbl 0465.35049 [9] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., Vol. 840, Springer-Verlag, Berlin, (1981). · Zbl 0456.35001 [10] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York-Heidelberg, (1972). · Zbl 0227.35001 [11] E. Nakaguchi; 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, 51-70 (2018) · Zbl 1391.35189 [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. · Zbl 0546.35003 [14] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Publishing Co., Amsterdam-New York, 1978. · Zbl 0387.46033 [15] F. B. Weissler, Local existence and nonexistence for semilinear parabolic equations in \(L^p\), Indiana Univ. Math. J., 29, 79-102 (1980) · Zbl 0443.35034 [16] A. Yagi, \(H^\infty\) Functional Calculus and Characterization of Domains of Fractional Powers, in Oper. Theory Adv. Appl., Vol. 187, 2008,217-235. · Zbl 1181.47012 [17] A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer-Verlag, Berlin, 2010. · Zbl 1190.35004 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.