On the existence of global solutions of evolution equations

• Autorzy: Medved M.
• Języki publikacji: EN

Abstrakty

EN

In this paper a sufficient condition for the existence of global solutions of evolution equations is proved. In the proof a modification of the Bihari type integral inequality to the case of a weakly singular nonlinear integral inequality is used. An application to a reaction-diffusion problem is given.

Słowa kluczowe

EN

integral inequality; evolution equation; mild solution; global solution; raction-diffusion equations

PL

równania całkowe; równania pierwiastkowe; globalne rozwiązania; nierówności integralne

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2004
• Tom: Vol. 37, nr 4
• Strony: 871--882
• Opis fizyczny: Bibliogr. 29 poz.

Twórcy

autor

Medved M.

• Department of Mathematical Analysis, Faculty of Mathematics, Physics and Informatics, Comenius University, Mlynská dolina, 842 48 Bratislava, Slovakia

Bibliografia

• [1] H. Amann, Global existence of solutions of quasilinear parabolic systems, J. Reine Angew. Math. 360 (1985), 47-83.
• [2] H. Amann, Dynamic theory of quasilinear parabolic systems: Global existence, Math. Z. 202 (1989), 219-250.
• [3] H. Amann, Global existence of a class of highly degenerate parabolic systems, Japan J. Indus. Appl. Math. 8 (1991), 143-159.
• [4] E. F. Beckenbach and R. Bellman, Inequalities, Springer, Berlin, Gottingen, Heidelberg, 1961.
• [5] J. A. Bihari, A generalization of a lemma of Bellman and its applications to uniqueness problems of differential equations, Acta Math. Acad. Sci. Hungar. 7 (1965), 81-94.
• [6] P. J. Bushell and W. Okrasinski, On the maximal interval of existence for solutions to some non-linear Volterra integral equations with convolution kernel, Bull. London Math. Soc. 28 (1995), 59-65.
• [7] A. Constantin, Solutions globales des equations différentielles perturbeés, C. R. Acad. Sci. Paris 320 (1995), 1319-1322.
• [8] A. Constantin and S. Peszat, Global existence of solutions of semilinear parabolic evolution equations, Differential and Integrad Equations 13 (1-3) (2000), 99-114.
• [9] M. Fila, Boundedness of global solutions of nonlinear differential equations, J. Differential Equations 28 (1992), 226-240.
• [10] M. Fila and H. A. Levine, On the boundedness of global slutions of abstract semilinear parabolic equations, J. Math. Anal. Appl 216 (1997), 654-666.
• [11] A. N. Filatov and L. V. Sharova, Integral Inequcdities and Theory of Nonlinear Oscillations, Nauka, Moscow 1976 (in Russian).
• [12] J. K. Hale, Asymptotic Behaviour of Dissipative Systems, in: Mathematical Surveys and Monographs, AMS 25, Providence 1988.
• [13] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin, Heidelberg, New York 1981.
• [14] K. Kirane and N. Tatar, Global existence and global stability of some semilinear problems, Archivum Math. (Brno) 36 (2000), 33-44.
• [15] A. A. Martyniuk and R. Gutowski, Integral Inequalities and Stability of Motion, Naukova Dumka, Kiev 1979 (in Russian).
• [16] A. A. Martyniuk, V. Lakshmikanthan and S. Leela, Motion Stability: The Method of Integral Inequalities, Naukova Dumka, Kiev 1977 (in Russian).
• [17] A. H. Martin, Global existence questions for reaction-diffusion systems, Pitman Res. Notes Math. 1 (1986), 169-177.
• [18] S. McKee and T. Tang, Integral inequalities in numerical analysis, Fasc. Mah. 308, 23 (1991), 67-76.
• [19] M. Medved, A new approach to an analysis of Henry type inequlities and their Bihari type versions, J. Math. Anal. Appl. 214 (1997), 349-366.
• [20] M. Medved, Singular integral inequalities and stability of semilinear parabolic equations, Archivum Math. (Brno) 34, 1 (1998), 183-190.
• [21] M. Medved, Nonlinear integral inequalities for functions in two and n independent variables, J. Inequal. Appl. 5 (2000), 1-22.
• [22] M. Medved, Nonlinear singular difference inequalities suitable for discretizations of parabolic equations, Demonstratio Math. 33, 3 (2000), 511-525.
• [23] M. Medved, Integral inequlities and global solutions of semilinear evolution equations, J. Math. Anal. Appl. 267 (2002), 643- 650.
• [24] M. Mizoguchi and E. Yanagida, Blow-up of solutions with sign-changes for semilinear diffusion equations, J. Math. Anal. Appl 204 (1996), 283-290.
• [25] K. Naito, Periodically reachable sets for nonlinear parabolic systems under periodic forcing, Yokohama Math. J. 43 (1995), 13-35.
• [26] B. G. Pachpatte, On some new inequalities in the theory of difference equtions, J. Math. Anal. Appl. 189 (1995), 128-144.
• [27] A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations, Springer, New York 1983.
• [28] H. Sano and N. Kunimatsu, Modified Gronwall's inequality and its application to stabilization problem for semilinear parabolic systems, Systems Control Lett. 22 (1994), 145-156.
• [29] N. E. Tatar, Exponential decey for a semilinear problem with memory, Arab. J. Math. Sc. 7, 1 (2001), 29-45.