Global existence for the wave equation with nonlinear by Vitillaro E. PDF

By Vitillaro E.

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Differential Equations 186 (2002) 259–298 294 Now we distinguish two cases: (i) either Kk p0 for all kAN; % kAN: % (ii) or Kk > 0 for kXk; In the first case Z F ðÁ; uk Þp0 ð143Þ for all kAN; G1 and then, by (138), 1 k 2 1 jju jj þ jjruk jj22 þ 2 t 2 2 Z 0 t QðÁ; Á; ukt Þukt pE k ð0Þ: ð144Þ By (140) we have E k ð0ÞpC0 for some C0 ¼ C0 ðu0 ; u1 Þ > 0: Hence by (109) and (144), for any T > 0; k 2 1 2 jjut jj2 þ 12 jjruk jj22 pC0 ; ð145Þ jjukt jjLm ðð0;TÞÂG1 Þ pcÀ1 7 C0 ; ð146Þ where c7 is the constant appearing in (109) for Y ¼ T: Then, by (108), jjQðÁ; Á; ukt ÞjjLm0 ðð0;TÞÂG1 Þ pC1 ð147Þ for some C1 ¼ C1 ðu0 ; u1 ; m; G1 ; c6 ðTÞ; c7 ðTÞÞ > 0 (here c6 is the constant appearing in (108) for Y ¼ T).

Control Optim. 28 (1990) 466–477.

8] G. Chen, A note on the boundary stabilization of the wave equation, SIAM J. Control Optim. 19 (1) (1981) 106–113. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, Toronto, London, 1955. E. Edwards, Functional Analysis, Theory and Applications, Holt, Rinehart and Winston, New York, 1965. [11] V. Georgiev, G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differential Equations 109 (1994) 295–308.

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Global existence for the wave equation with nonlinear boundary damping and source terms by Vitillaro E.


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