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