Some new results on global nonexistence and blow-up for evolution problems with positive initial energy

Rendiconti dell'Istituto di Matematica dell'Universita di Trieste

Volumn 31, Issue , 2000, Pages 245-275

  Vitillaro, Enzo ^a  

^a UNIVERSITY OF PERUGIA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001470666     PISSN: 00494704     EISSN: 24648728     Source Type: Journal    
DOI: None     Document Type: Article

References (48)

1. R. A ADAMS, Sobolev spaces, Academic Press, New York, 1975
2. R. A AMBROSETTI, Critical points and nonlinear variational problems, Supplement au Bulletin de la Societe Mathematique de France, Memoires 120 (1992), no. 49
3. J. M. BALL, Finite time blow-up in nonlinear problems, Nonlinear evolution equations, Proc. Symp. (Madison/Wis.), 1978, pp. 189-205
4. M. M. CAVALCANTI, V. N. DOMINGOS CAVALCANTI, AND J. A SORIANO, Existence and boundary stabiization of a nonlinear hyperbolic equation with time dependent coefcients, Electron. J. Differential Equations 8 (1998), 1-21
5. G. CHEN, Control and stabilization for the wave equation in a bounded domain, SIAM J. Control Optim. 17 (1979), 66-81
6. G. CHEN, Energy decay estimates and exact bounday value control lability for the wave equation in a bounded domain, J. Math. Pures Appl. 58 (1979), 249-273
7. G. CHEN, A note on the boundary stabilization of the wave equation, SIAM J. Control Optim. 19 (1981), no. 1, 106-113
8. G. CHEN, Control and stabilization for the wave equation in a bounded domain, part II, SIAM J. Control Optim. 19 (1981), 114-122
9. J. FILO, On solutions of a perturbed fast diffusion equation, Apl. Mat. 32 (1987), no. 5, 364-380
10. 1+α, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 13 (1966), 109-124
11. H. FUJITA, On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations, Proc. Symp. Pure Math. XVIII Nonlinear Functional Analysis (Provedence, RI.), American Mathematical Society, 1968, pp. 105-124
12. β+1, Differentsial'nye Uravneniya 17 (1981), no. 5, 836-842, In Russian: English translation, Differential Equations, 17 (1981), 551-555
13. V. GEORGIEV AND G. TODOROVA, Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differential Equations 109 (1994), 295-308
14. R. T. GLASSEY, Blow-up theorems for nonlinear wave equations, Math. Z. 12 (1973), 183-203
15. R. IKEHATA, Some remarks on the wave equations with nonlinear damping and source terms, Nonlnear Anal. 27 (1996), no. 10, 1165-1175
16. H. ISHII, Asymptotic stability and blowing up of solutions of som nonlinear equations, J. Differential Equations 26 (1977), 291-319
17. K. JORGENS, Nonlinear wave equations, Lectures Notes, University of Colorado, 1970
18. S. KAPLAN, On the growth of solutions of quasilinear parabol equations, Comm. Pure Appl. Math. 16 (1963), 305-330
19. J. B. KELLER, On solutions of nonlinear wave equations, Comm. Pure Appl. Math. 10 (1957), 523-530
20. H. A. KNOPS, R. J.AND LEVINE AND L. E. PAYNE, Nonexistence, instability, and growth theorems for solutions of a clss of abstract nonlinear equations with applications to nonlinear elstodynamic, Arch. Rational Mech. Anal. 55 (1974), no. 1, 52-72
21. V. KOMORKIN AND E. ZUAZUA, A direct method for boundary stabi lization of the wave equation, J. Math. Pures Appl. 69 (1990), 33-54
22. J. LAGNESE, Decay of solutions of wave equations in a bounded region with boudary dissipation, J. Differential Equations 50 (1983), 163-182
23. J. LAGNESE, Note on boundary stabilization of wave equations, SIAM J. Control Optim 26 (1988), 1250-1256
24. I. LASIECKA AND D. TATARU, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differ ential Integral Equation 6 (1993), no. 3, 507-533
25. t =-Au + F(u))., Arch. Rational Mech. Anal. 51 (1973), 371-386
26. tt =-Au + F(u)), Trans. Amer. Math. Soc. 192 (1974), 1-21
27. H. LEVINE, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal. 5 (1974), 138-146
28. H. LEVINE, Nonexistence of global solutions of nonlinear wave equations, Improperly ped boundary value problems, Pitman, London, 1975, pp. 94-104
29. EL A. LEVINE, S. R. PARK, AND J. SERRIN, Global existence and nonexistence theorems for quasilinear evolution equations of formal parabolic type, J. Differential Equations 142 (1998), 212-229
30. EL A. LEVINE, P. Pucci, AND J. SERRIN, Some remarks on global nonexistence for nonautonomous abstract evolution equations, Har monic Analysis and Nonlinear Differential Equations (M. L. Lapidus, L. H. Harper, and A. J. Rumbos, eds.), Contemp. Math., vol. 208, American Mathematical Society, 1997, pp. 253-263
31. H. A. LEVINE AND P. E. SACKS, Some existence and nonexistence theorems for solutions of degenerate parabolc equations, J. Differential Equations 52 (1984), 135-161
32. H. A. LEVINE AND J. SERRIN, Gobal nonexistence theorems for quasilinear evolutions equations with dissipaton, Arch. Rational Mech. Anal. 17 (1997), 341-361
33. H. A. LEVINE AND A. SMITH, A potential well theory for the wave equation with a nonlinear boundary condition, J. Reine Angew. Math. 34 (1987), 1-23
34. H. A. LEVINE AND G. TODOROVA, Blow up of solutions of the Cauchy problem for a wave equation with nonlinear damping and source term and large initial energy, preprint
35. M. NAKAO, Existence, nonexistence and some asymptotic behavior of global solutions of a nonlinear degenerate parabol equation, Math. Rep. Kyushu Univ. 14 (1983), 1-21
36. L. E. PAYNE AND D. H. SATTINGER, Saddle points and instability of nonlinear hyperbol equations, Israel J. Math. 22 (1975), 273-303
37. P. Pucci AND J. SERRIN, Global nonexistence for abstract evolution equations with positive initial energy, J. Differential Equations 150 (150), no. 1, 203-214
38. J. QUINN AND D. L. RUSSEL, Asymptotic stability and energy decay for solutions of hyperbol equation, Proc. Roy. Soc. Edinburgh Sect. A 77 (1977), 97-127
39. A. SAMARSKII, V. A. GALAKTIONOV, S. P. KURDYUMOV, AND . P. MIKHAILOV, Bow-up in quasilinear parabolic equations, de Gruyter expositions in athematics, vol 19, Walter de ruyter, Berln-New York, 1995
40. D. H. SATTINGER, On global solutions of nonlinear hyperbol equations, Arch. Rational Mech. 30 (1968), 148-172
41. D.H. SATTINGER, Stability of nonlinear hyperbol equations, Arch. Rational Mech. Anal. 28 (1968), 226-244
42. J. SERRIN, G. TODOROVA, AND E. VITILLARO, Local existence for a nonlinear wave equation with nonlinear damping and source terms, preprint
43. M. TSUTSUMI, Existence and nonexistence of global solutions for nonlinear parabol equations, Publ. Res. Inst. Math. Sci. 8 (1972), 211 229
44. M. TSUTSUMI, On solutions of semilinear differential equations i Eilbert space, Math. Japon. 17 (1972), 173-193
45. E. VITILLARO, Blow-up for the porous media equation with source term and positive initial energy, to appear on J. Math. Anal. Appl
46. E. VITILLARO, Global nonexistence theorems for a class of evolution equations with dissipation and applcation, Arch. Rational Mech. Anal 149 (1999), 155-182
47. W. P. ZIEMER, Weakly differentiable functions, Graduate Text athematics, vol. 120, Springer, New York, 1989
48. E. ZUAZUA, Uniform stabilization of the wave equation by nonlnear boundary feedback, SIAM J. Control Opti. 28 (1990), 466-477