Search

In this paper we consider the Boussinesq-type equation associated with logarithmic nonlinear term and initial boundary value conditions. By using potential well method combined with the logarithmic Sobolev inequality, we obtain the existence of global solution. At last we show that the L 2 -norm of the solution will grow up as an exponential function as time goes to infinity under some suitable conditions for initial data. The result shows that the polynomial nonlinearity is a critical condition of blow-up in finite time for the solutions of Boussinesq equations. [ABSTRACT FROM AUTHOR]

In this paper we consider the Cauchy problem of two-dimensional generalized Boussinesq-type equation u t t − Δ u − Δ u t t + Δ 2 u + Δ f ( u ) = 0 . Under the assumption that f ( u ) is a function with exponential growth at infinity and under some assumptions on the initial data, we prove the existence and nonexistence of global weak solution. There are very few works on Boussinesq equation with nonlinear exponential growth term by potential well theory. [ABSTRACT FROM AUTHOR]

In this paper, we consider the optimization problems with k -th order stochastic dominance constraint on the objective function of the two-stage stochastic programs with full random quadratic recourse. By establishing the Lipschitz continuity of the feasible set mapping under some pseudo-metric, we show the Lipschitz continuity of the optimal value function and the upper semicontinuity of the optimal solution mapping of the problem. Furthermore, by the Hölder continuity of parameterized ambiguity set under the pseudo-metric, we demonstrate the quantitative stability results of the feasible set mapping, the optimal value function and the optimal solution mapping of the corresponding distributionally robust problem. [ABSTRACT FROM AUTHOR]

In this paper, we study a distributionally robust risk optimization (DRRO) problem where the information on the probability distribution of the underlying random variables is incomplete. But it is possible to use partial information to construct an ambiguity set of probability distributions. In some cases, decision vector x may have a direct impact on the likelihood of the underlying random events that occur after the decision is taken, which motivates us to propose an ambiguity set to be parametric and decision-dependent. To conduct quantitative stability analysis of the optimal value function and the optimal solution mapping of the DRRO problem, we derive error bounds results for the parametrized ambiguity set under the total variation metric and investigate Lipschitz continuity of the objective function of the DRRO problem under some conditions. As an application, we demonstrate that the two-stage stochastic linear semi-definite programs satisfy these conditions and then apply results obtained to it. [ABSTRACT FROM AUTHOR]