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Theoretical and numerical analysis of a class of quasilinear elliptic equations
A semilinear heat equation with initial data in negative Sobolev spaces
Kusauchi Yamashina 53-7, Kyotanabe City, Kyoto, Japan |
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.
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. 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. |
[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. 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. |
[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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