TAILIEUCHUNG - Existence, global nonexistence, and asymptotic behavior of solutions for the Cauchy problem of a multidimensional generalized damped Boussinesq-type equation
We consider the existence, both locally and globally in time, the global nonexistence, and the asymptotic behavior of solutions for the Cauchy problem of a multidimensional generalized Boussinesq-type equation with a damping term. | Turk J Math (2014) 38: 706 – 727 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Existence, global nonexistence, and asymptotic behavior of solutions for the Cauchy problem of a multidimensional generalized damped Boussinesq-type equation ˙ ¸ KIN, ˙ Erhan PIS Necat POLAT∗ Dicle University, Department of Mathematics, 21280 Diyarbakır, Turkey Received: • Accepted: • Published Online: • Printed: Abstract: We consider the existence, both locally and globally in time, the global nonexistence, and the asymptotic behavior of solutions for the Cauchy problem of a multidimensional generalized Boussinesq-type equation with a damping term. Key words: Existence, global nonexistence, asymptotic behavior, Boussinesq equations, damping term 1. Introduction In this paper, we study the Cauchy problem of the generalized multidimensional Boussinesq-type equation with a damping term utt − △u − a △ utt + △2 u + △2 utt − k △ ut = △f (u) , u (x, 0) = u0 (x) , ut (x, 0) = u1 (x) , (x, t) ∈ Rn × (0, +∞) , x ∈ Rn , () () where u (x, t) denotes the unknown function, f (s) is the given nonlinear function, u0 (x) and u1 (x) are the given initial value functions, k is a constant, the subscript t indicates the partial derivative with respect to t, n is the dimension of space variable x, and △ denotes the Laplace operator in Rn . The effects of small nonlinearity and dispersion are taken into consideration in the derivation of Boussinesq equations, but in many real situations, damping effects are compared in strength to the nonlinear and dispersive ones. Therefore, the damped Boussinesq equation is considered as well: ( ) utt − 2butxx = −αuxxxx + uxx + β u2 xx , () where utxx is the damping term, a, b = const > 0, and β = const ∈ R (see [6] and references therein). Varlamov [12] investigated the long-time behavior of solutions to initial value, .
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