A potential well theory for the wave equation with nonlinear source and boundary damping terms

Abstract

Abstract. The paper deals with local existence, blow-up and global existence for the solutions of a wave equation with an internal nonlinear source and a nonlinear boundary damping. The typical problem studied is; 1Þ Â À 1 ; uð0; xÞ ¼ u 0 ðxÞ; u t ð0; xÞ ¼ u 1 ðxÞ on ;where & R n (n ! 1) is a regular and bounded domain,1 Þ, ! 0, and the initial data are in the energy space. The results proved extend the potential well theory, which is well known when the nonlinear damping acts in the interior of , to this problem.

Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping–source interaction

“…The case of sole interior source f (s) = |s| k−1 k with subcritical exponent k < n n−2 , h = 0 and nonlinear boundary damping g(s) was treated in [7]. In [7] uniform decay rates similar to these obtained earlier in [29] were derived for small initial data taken from potential well constructed in [33]. Thus the results of Theorem 2.6, which does not assume any a priori structure of the sources and allows for a completely arbitrary behaviour of the damping at the origin, subsume and significantly extend the results obtained in the prior literature.…”

“…And, in fact, this does happen if the nonlinear functions are of special form and the initial data are taken from a special set-so-called potential well. In the case when one source (be it either boundary or interior) is active and the damping is polynomial, the potential well theory has been developed in [33,34] and references therein. We shall show that a similar construction can be performed for two competing sources (boundary and interior), and without assuming any (polynomial) structure on the damping.…”

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.…”

Higher Order Wave Equation with Logarithmic Source Term

“…Hiramatsu [4] when studying the dynamics of Q-balls in theoretical physics. There is extensive literature on logarithmic equation used by Enzo Vitillaro [2] and Gorka P. [3], nonlinearity of waves used by Bialynicki-Birula, I., Mycielski, J. [6] , existence of solution used by Han X.…”