Qualitative analysis of a nonlinear wave equation

Global solutions and finite time blow up for damped semilinear wave equations

“…damping terms such as |u t | m−2 u t with m > 2 (often enlightening the interaction that pops up with the corresponding power source |u| p−2 u). We refer the reader to [6,9,13,25,26,29] and to the references therein. In analogy with these extensions, one could wonder whether it is possible to obtain some results for a nonlinear strong damping such as − m u t (the m-Laplacian operator).…”

“…Cazenave [4] proved boundedness of global solutions to (1.1) for ω = µ = 0 while Esquivel-Avila [5] recovered the same result for ω = 0 and µ > 0 and showed that this property may fail in presence of a nonlinear dissipation term (cf. [6,Theorem 3.4]). In Theorems 3.4 and 3.6, by exploiting an argument different from the one devised in [4,5], we prove that any global solution of (1.1) is bounded whenever ω and µ fulfill (1.3).…”

On existence, uniform decay rates and blow up for solutions of the 2-D wave equation with exponential source

“…The majority of the works in the literature makes use of the above theory when g possesses polynomial growth. See, for instance the following works: [1,5,[7][8][9][10][11][12][14][15][16][18][19][20][25][26][27][28][29][30][31][32][34][35][36] and a long list of references therein.…”

Global Existence and Blowing Up of Solutions for Some Non-linear Wave Equations

“…However, we mention that, by exploiting a completely different method, the existence of solutions with arbitrarily high initial energy has been also obtained in [11] for weakly damped wave equations on the whole R n . Cazenave [2] proved bounded-ness of global solutions for ω = μ = 0, while Esquivel-Avila [5,6] recovered the same result for ω = 0 and μ > 0 and showed that this property may fail in the presence of a nonlinear dissipation term.…”

“…where is the map defined in (6). The proof in the case ω = 0 (and μ ≥ 0) is similar and follows by obvious modifications of inequalities (20) and (24).…”