Stability for semilinear parabolic equations with decaying potentials in $\mathbb {R}^n$ and dynamical approach to the existence of ground states

Souplet, Philippe; Zhang, Qi S.

Stability for semilinear parabolic equations with decaying potentials in

ℝ^{n}

and dynamical approach to the existence of ground states

Souplet, Philippe ; Zhang, Qi S.

Annales de l'I.H.P. Analyse non linéaire, Tome 19 (2002) no. 5, pp. 683-703.

Zbl

@article{AIHPC_2002__19_5_683_0,
     author = {Souplet, Philippe and Zhang, Qi S.},
     title = {Stability for semilinear parabolic equations with decaying potentials in $\mathbb {R}^n$ and dynamical approach to the existence of ground states},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {683--703},
     publisher = {Elsevier},
     volume = {19},
     number = {5},
     year = {2002},
     zbl = {1017.35033},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2002__19_5_683_0/}
}

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JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2002
SP  - 683
EP  - 703
VL  - 19
IS  - 5
PB  - Elsevier
UR  - http://www.numdam.org/item/AIHPC_2002__19_5_683_0/
LA  - en
ID  - AIHPC_2002__19_5_683_0
ER  -

%0 Journal Article
%A Souplet, Philippe
%A Zhang, Qi S.
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%J Annales de l'I.H.P. Analyse non linéaire
%D 2002
%P 683-703
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%G en
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Souplet, Philippe; Zhang, Qi S. Stability for semilinear parabolic equations with decaying potentials in $\mathbb {R}^n$ and dynamical approach to the existence of ground states. Annales de l'I.H.P. Analyse non linéaire, Tome 19 (2002) no. 5, pp. 683-703. http://www.numdam.org/item/AIHPC_2002__19_5_683_0/

Bibliographie
Cité par

[1] Bahri A., Lions P.L., On the existence of a positive solution of semilinear elliptic equations in unbounded domains, Ann. Inst. Henri Poincaré, Anal. non linéaire 14 (1997) 365-413. | Numdam | MR | Zbl

[2] Berestycki H., Capuzzo Dolcetta I., Nirenberg L., Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Topol. Methods Nonlinear Anal. 4 (1994) 59-78. | MR | Zbl

[3] Berestycki H., Lions P.L., Nonlinear scalar field equations I, existence of a ground state, Arch. Rat. Mech. Anal. 82 (1983) 313-346. | MR | Zbl

[4] Berestycki H., Lions P.L., Nonlinear scalar field equations II, existence of infinitely many solutions, Arch. Rat. Mech. Anal. 82 (1983) 347-376. | MR | Zbl

[5] Cazenave T., Lions P.L., Solutions globales d'équations de la chaleur semilinéaires, Comm. Partial Differential Equations 9 (1984) 955-978. | MR | Zbl

[6] Cazenave T., Weissler F.B., Asymptotically self-similar global solutions of the nonlinear Schrödinger and heat equations, Math. Z. 228 (1998) 83-120. | MR | Zbl

[7] Deng K., Levine H.A., The role of critical exponents in blowup theorems, the sequel, J. Math. Anal. Appl. 243 (2000) 85-126. | MR | Zbl

[8] Ding W.Y., Ni W.M., On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Rat. Mech. Anal. 91 (1986) 283-308. | MR | Zbl

[9] Feireisl E., Petzeltova H., Convergence to a ground state as a threshold phenomenon in nonlinear parabolic equations, Differential Integral Equations 10 (1997) 181-196. | MR | Zbl

[10] Fila M., Souplet Ph., Weissler F.B., Linear and nonlinear heat equations in L^q_δ spaces and universal bounds for global solutions, Math. Annalen 320 (2001) 87-113. | Zbl

[11] Fujita H., On the blowing up of solutions of the Cauchy problem for u_t=Δu+u^1+α, J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966) 109-124. | Zbl

[12] Gidas B., Spruck J., Global and local behaviour of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981) 525-598. | MR | Zbl

[13] Giga Y., A bound for global solutions of semilinear heat equations, Comm. Math. Phys. 103 (1986) 415-421. | MR | Zbl

[14] Haraux A., Weissler F.B., Non-uniqueness for a semilinear initial value problem, Indiana Univ. Math. J. 31 (1982) 167-189. | MR | Zbl

[15] Kato T., Strong L^p-solutions of the Navier-Stokes equation in R^m, with applications to weak solutions, Math. Z. 187 (1984) 471-480. | MR | Zbl

[16] Kuzin I., Pohozaev S., Entire Solutions of Semilinear Elliptic Equations, Birkhäuser, 1997. | MR | Zbl

[17] Levine H.A., Some nonexistence and instability theorems for solutions of formally parabolic equations of the form Pu_t=−Au+F(u), Arch. Rat. Mech. Anal. 51 (1973) 371-386. | Zbl

[18] Levine H.A., The role of critical exponents in blow up theorems, SIAM Rev. 32 (1990) 262-288. | MR | Zbl

[19] Lieberman G., Second Order Parabolic Differential Equations, World Scientific, Singapore, 1996. | MR | Zbl

[20] Lions P.L., The concentration compactness principle in the calculus of variation. The locally compact case, part I, II, Ann. Inst. Henri Poincaré, Anal. non linéaire 1 (1984) 109-145, 223-283. | Numdam | Zbl

[21] Moser J., A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964) 101-134. | MR | Zbl

[22] Murata M., Structure of positive solutions to (−Δ+V)u=0 in R^m, Duke Math. J. 53 (1986) 869-943. | Zbl

[23] Ni W.M., Sacks P., Tavantzis J., On the asymptotic behavior of solutions of certain quasilinear parabolic equations, J. Differential Equations 54 (1984) 97-120. | MR | Zbl

[24] Noussair E., Swanson C.A., Existence theorems for generalized Klein Gordon equations, Bull. Amer. Math. Soc. 31 (1983) 333-336. | MR | Zbl

[25] Quittner P., Universal bound for global positive solutions of a superlinear parabolic problem, Math. Annalen 320 (2001) 299-305. | MR | Zbl

[26] Souplet Ph., Sur l'asymptotique des solutions globales pour une équation de la chaleur semi-linéaire dans des domaines non bornés, C. R. Acad. Sci. Paris 323 (1996) 877-882. | MR | Zbl

[27] Souplet Ph., Geometry of unbounded domains, Poincaré inequalities and stability in semilinear parabolic equations, Comm. Partial Differential Equations 24 (1999) 951-973. | MR | Zbl

[28] Souplet Ph., Zhang Q.S., Existence of ground states for semilinear elliptic equations with decaying mass: a parabolic approach, C. R. Acad. Sci. Paris 332 (2001) 515-520. | MR | Zbl

[29] Weissler F.B., Semilinear evolution equations in Banach spaces, J. Funct. Anal. 32 (1979) 277-296. | MR | Zbl

[30] Weissler F.B., Local existence and nonexistence for semilinear parabolic equations in L^p, Indiana Univ. Math. J. 29 (1980) 79-102. | MR | Zbl

[31] Weissler F.B., Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math. 38 (1981) 29-40. | MR | Zbl

[32] Zhang Q.S., Large time behavior of Schrödinger heat kernels and applications, Comm. Math. Phys. 210 (2000) 371-398. | MR | Zbl

[33] Zhang Q.S., Semilinear parabolic equations on manifolds and applications to the non-compact Yamabe problem, Electron. J. Differential Equations 46 (2000) 1-30. | MR | Zbl