This paper is concerned with the modified Wigner (respectively, Wigner–Fokker–Planck) Poisson equation. The quantum mechanical model describes the transport of charged particles under the influence of the modified Poisson potential field without (respectively, with) the collision operator. Existence and uniqueness of a global mild solution to the initial boundary value problem in one dimension are established on a weighted $L^{2}$ -space. The main difficulties are to derive a priori estimates on the modified Poisson equation and prove the Lipschitz properties of the appropriate potential term.

This paper is devoted to investigating formation of singularities for solutions to semilinear Moore-Gibson-Thompson equations with power type nonlinearity $ |u|^{p} $, derivative type nonlinearity $ |u_{t}|^{p} $ and combined type nonlinearities $ |u_{t}|^{p}+|u|^{q} $ in the case of single equation, combined type nonlinearities $ |v_{t}|^{p_{1}}+|v|^{q_{1}} $, $ |u_{t}|^{p_{2}}+|u|^{q_{2}} $, combined and power type nonlinearities $ |v_{t}|^{p_{1}}+|v|^{q_{1}} $, $ |u|^{q_{2}} $, combined and derivative type nonlinearities $ |v_{t}|^{p_{1}}+|v|^{q_{1}} $, $ |u_{t}|^{p_{2}} $ in the case of coupled system, respectively. More precisely, blow-up results of solutions to problems in the sub-critical and critical cases are derived by applying test function technique. Moreover, upper bound lifespan estimates of solutions to the coupled systems are investigated. The main new contribution is that lifespan estimates of solutions are associated with the well-known Strauss exponent and Glassey exponent.

This article studies the Cauchy problem for the damped wave equation with nonlinear memory. For a noncompactly supported initial data with small energy, global existence and asymptotic behaviour of solutions are obtained when 1 ≤ n ≤ 3. This result generalized the previous result by Fino [Critical exponent for damped wave equations with nonlinear memory, Nonlinear Anal. 74 (2011), pp. 5495–5505], which dealt with the solution with compactly supported initial data.

An interaction equations of the complex scalar nucleon field and real scalar meson field is considered. we show that the Cauchy problem of the Klein-Gordon-Schrödinger system $ iu_{t}+u_{x x}=-uv, $ $ v_{t t}-v_{x x}+v=|u|^2,$ $u(0, x)= u_0(x), v(0, x)= v_0(x), v_t(0, x)= v_1(x)$ is locally well-posed for weak initial data $(u_0, v_0, v_1)\in H^s\times H^{s-1/2}\times H^{s-3/2}$ with $s\geq 0$. We use the analogous method for estimate the nonlinear couple terms developed by Bourgain and refined by Kenig, Ponce and Vega.

This paper is concerned with the Cauchy problem of the nonlinear fourth-order Schrödinger equation on $ \mathbb{R}^{n} $, with the nonlinearity of Hartree-type $ (|\cdot|^{-\gamma}\ast|u|^{2})u $. The existence of a global solution to the Cauchy problem is established when initial data belongs to $ L^{p}(p<2) $. Moreover, the solution is given by using a data-decomposition argument, two kinds of generalized Strichartz estimates, and the interpolation theorem.