On Solvability of the Dissipative Kirchhoff Equation With Nonlinear Boundary Damping

Abstract

Abstract. In this paper, we prove the global existence and uniqueness of the dissipative Kirchhoff equationwith nonlinear boundary damping by Galerkin approximation benefited from the ideas of Zhang et al. [33]. Furthermore,we overcome some difficulties due to the presence of nonlinear terms M ( ∇u 2 ) and g(ut) by introducing a new variables and we can transform the boundary value problem into an equivalent one with zero initial data by argument of compacity and monotonicity.

Existence and asymptotic behaviour of solutions for the wave equation with a time varying delay and logarithmic nonlinearity

“…In this section, we are going to prove Theorem 2.11. That is, benefit from Lyapunov function method inspired by the contributions (see, for instance, [17], [18], [31], [32]), we establish the asymptotic behaviour of the energy. Moreover, we prove that the asymptotic behaviour is optimal.…”

“…Moreover, we relax the constraints on function G 0 which appears in Section 4 to obtain two types of decay results under different assumptions on G. More precisely, we construct polynomial decay result by introducing an auxiliary functional in one case. In another case, we obtain the exponential decay result by the perturbed energy method (see, for instance, [17,18,31,32]).…”

Well-Posedness and Blow-Up of Solutions for a Variable Exponent Nonlinear Petrovsky Equation

“…They also proved the finite time blow-up of solutions. Moreover, numerous researchers have studied the mathematical behavior of equations using the Faedo-Galerkin and the perturbed energy method [10][11][12][13][14].…”

Optimal control of a viscous generalized θ-type dispersive equation with weak dissipation

“…is called a weak solution to (2.1), if Next, by using the Galerkin method [27,51,52] and a priori estimates, one can obtain the following theorem, which ensures the existence of a unique weak solution to (2.1). Now, we are in position to state the main result in this section.…”