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Reine Angew. Math. 374 (1987) 1–23. L. Lions, Quelques me´thodes de re´solutions des proble`mes aux limites non line´aires, Dunod, Paris, 1969. [21] P. Pucci, J. Serrin, Global nonexistence for abstract evolution equations with positive initial energy, J. Differential Equations 150 (1) (1998) 203–214. L. A. Strauss, On some nonlinear evolution equations, Bull. Soc. Math. France 93 (1965) 43–96. L. Royden, Real Analysis, MacMillan, New York, 1963. [24] I. Segal, Nonlinear Semigroups, Ann. of Math.

Differential Equations 16 (1974) 319–334. A. Levine, J. Serrin, Global nonexistence theorems for quasilinear evolutions equations with dissipation, Arch. Rational Mech. Anal. 137 (1997) 341–361. A. Levine, A. Smith, A potential well theory for the wave equation with a nonlinear boundary condition, J. Reine Angew. Math. 374 (1987) 1–23. L. Lions, Quelques me´thodes de re´solutions des proble`mes aux limites non line´aires, Dunod, Paris, 1969. [21] P. Pucci, J. Serrin, Global nonexistence for abstract evolution equations with positive initial energy, J.

E. Vitillaro / J. Differential Equations 186 (2002) 259–298 288 for 0ptpT: Using (F1) and (Q3) we have þ 12 jjrwjj22 þ c5 jjwt jjm Lm ðð0;TÞÂG1 Þ Z TZ pc8 ½ju À ujð1 % þ jujpÀ2 þ juj % pÀ2 Þ þ ju À uj % qÀ1 Šjwt j; 2 1 2 jjwt jj2 ð122Þ G1 0 when mX2; while 0 þ 12 jjrwjj22 þ c5 jjjvt jmÀ2 vt À jv%t jmÀ2 v%t jjm Lm0 ðð0;TÞÂG1 Þ Z TZ pc8 ½ju À ujð1 % þ jujpÀ2 þ juj % pÀ2 Þ þ ju À uj % qÀ1 Šjwt j; 2 1 2 jjwt jj2 ð123Þ G1 0 when 1omo2: To estimate the right-hand side of (122) and (123) we first note that to estimate Z TZ I2 :¼ ju À ujð1 % þ jujpÀ2 þ juj % pÀ2 Þjwt j; G1 0 since G1 is bounded, we can use the same arguments employed for term I1 in the proof of Theorem 1, and prove (remember that RX1) that 0 I2 p k2 ð1 þ 2RpÀ2 ÞðR2=ðmÀ1Þ T 1=m þ RðpÀ1Þ=ðmÀ1Þ TÞ jju À ujj % LN ð0;T;Lr0 ðG1 ÞÞ ; ð124Þ generalizing (78).

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


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