In this note, we calculate the boundary of movable cones and nef cones of the generalized Kummer 4-fold $\mathrm{Km}^2(A)$ attached to an abelian surface $A$ with $\mathrm{rk} \mathrm{NS}(A) = 1$.

We analyze the asymptotic behavior of solutions to wave equations with strong damping terms in $\textbf{R}^n$ $(n\ge1)$, $$ u_{tt}-\Delta u-\Delta u_t=0, \qquad u(0,x)=u_0(x), \quad u_t(0,x)=u_1(x). $$ If the initial data belong to suitable weighted $L^1$ spaces, lower bounds for the difference between the solutions and the leading terms in the Fourier space are obtained, which implies the optimality of expanding methods and some estimates proposed in [13] and in this paper.

We consider the long time behavior of solutions to the initial value problem for the ``complex-valued'' cubic nonlinear Klein-Gordon equation (NLKG) in one space dimension. In [12], Sunagawa derived the $L^{\infty}$ decay estimate of solutions to (NLKG). In this note, we obtain the large time asymptotic profile of solutions to (NLKG).

In this paper, we deal with the chemotaxis-haptotaxis system with logistic source under homogeneous Neumann boundary conditions in a bounded convex domain with smooth boundary. Under appropriate regularity assumptions on the initial data, by $L^p$-estimate techniques, we show that the system possesses at least one global and bounded weak solution. Our results generalize and improve previous results.

Let $\Gamma$ be a discrete group satisfying the approximation property (AP). Let $X$, $Y$ be $\Gamma$-spaces and $\pi \colon Y \to X$ be a proper factor map which is injective on the non-free part. We prove the one-to-one correspondence between intermediate C$^*$-algebras of $C_0(X) \rtimes_r \Gamma \subset C_0(Y) \rtimes_r \Gamma$ and intermediate $\Gamma$-${\rm C}^\ast$-algebras of $C_0(X) \subset C_0(Y)$. This is a generalization of Suzuki's theorem that proves the statement for free actions.

We consider the Riemann problem for fluids of van der Waals type with phase transitions involving nonclassical shocks. The model is elliptic-hyperbolic and the pressure function admits two inflection points. First, a unique classical Riemann solver is constructed, which is based on rarefaction waves, classical shocks and zero-speed shocks. Second, we investigate nonclassical Riemann solvers, which involve nonclassical shocks. Nonclassical shocks are shock waves which violate the Liu entropy condition and satisfy a kinetic relation. It can be shown that then two wave curves always intersect either once or twice at different phases. Consequently, the Riemann problem always admit one or two solutions in the class of classical and nonclassical shocks, zero-speed shocks, and rarefaction waves.