We consider the Cauchy problem $$ u_t+\big[F(u)\big]_x=0,\qquad u(0,x)=\bar u(x)\tag{*} $$ for a nonlinear $n\times n$ system of conservation laws with coinciding shock and rarefaction curves. Assuming the existence of a coordinates system made of Riemann invariants, we prove the existence of a weak solution of $(*)$ that depends in a Lipschitz continuous way on the initial data, in the class of functions with arbitrarily large but bounded total variation.

We establish a uniqueness and existence theorem for bounded-from-below viscosity solutions of Hamilton-Jacobi equations of the form $u+H(Du)=f$ in $\mathbb{R}^N$. More precisely, we show that there is a unique solution $u$ such that $u^-$ grows at most linearly, when $f^-$ behaves analogously and when $H$ is convex and nonlinear. We sharpen this result when the behavior of the Hamiltonian is known at infinity. We also discuss some extensions to more general Hamiltonians and to the Dirichlet problem in an unbounded open set.

The Hamilton-Jacobi equation with concave Hamiltonians and degenerate coefficients is treated. A new definition of viscosity solutions is proposed to deal with lower semicontinuous solutions. Under the setting, the comparison principle for lower semicontinuous viscosity solutions is established. The justification of our definition in applications is shown.

We study the asymptotic behavior near finite blow-up time $t = T$ of the solutions to the one-dimensional degenerate quasilinear parabolic equation $$ u_t = (u^\sigma u_x)_x + u^\beta \quad \text{in} \quad \mathbb{R} \times (0,T) ; \quad \sigma>0, \,\, 1<\beta<\sigma+1, $$ with bounded, nonnegative, compactly supported initial data. This parameter range corresponds to global blow-up where $ u(x,t) \to \infty$ as $t \to T^-$ for any $x \in \mathbb{R}$. We prove that the rescaled function $$ f(\xi,t) = (T-t)^{1/(\beta-1)} u(\xi(T-t)^m,t), \quad m = \frac {\beta-(\sigma+1)}{2(\beta-1)} < 0, $$ converges uniformly as $t \to T$ to a unique compactly supported, symmetric self-similar profile $\theta(\xi) \ge 0$ satisfying a nonlinear ordinary differential equation. The proof is based on the intersections comparison (the Sturmian argument) with a two-parametric family of self-similar solutions.

We prove that for small initial data there exist unique global solutions to a certain special class of nonlinear Klein-Gordon equations with cubic nonlinearity in one space dimension, which asymptotically approach the free solution of the linear Klein-Gordon equation as $t \to + \infty$.

We provide a general framework of stationary scattering theory for one-dimensional quantum systems with nontrivial spatial asymptotics. As a byproduct we characterize reflectionless potentials in terms of spectral multiplicities and properties of the diagonal Green's function of the underlying Shrödinger operator. Moreover, we prove that single (Crum-Darboux) and double commutation methods to insert eigenvalues into spectral gaps of a given background Shrödinger operator produce reflectionless potentials (i.e., solitons) if and only if the background potential is reflectionless. Possible applications of our formalism include impurity (defect) scattering in (half-) crystals and charge transport in mesoscopic quantum-interference devices.

Convergence to a fixed point for every positive semi-orbit of a monotone discrete-time dynamical system in a strongly ordered Banach space is investigated. The dynamical system is generated by a compact continuous self-mapping $T\colon [a,b]\to [a,b]$ of a closed order interval $[a,b]\overset{def}{=}(a + V_+)\cap (b - V_+)$ in an ordered Banach space $V$, where the positive cone $V_+$ of $V$ has nonempty interior $\overset{o}{V}_+ = \rm{Int} (V_+)$, and \hbox{$a,b\in V$} with $b-a\in \overset{o}{V}_+$. The mapping $T$ is strongly monotone on the open order interval $[[a,b]]\overset{def}{=} (a + \overset{o}{V}_+)\cap (b - \overset{o}{V}_+)$. Finally, assume that (i) the fixed points of $T$ contained in $[[a,b]]$ form a totally ordered set; (ii)for every nonempty compact set ${\cal K}\subset [[a,b]]$ of fixed points of $T$, $\min {\cal K}$ is lower Ljapunov stable or $\max {\cal K}$ is upper Ljapunov stable; (iii) for $n=1,2,\dots$, each fixed point $p\in [a,b]\setminus [[a,b]]$ of $T^n$, $a\neq p\neq b$, is ejective; and (iv) if dim $(V) < \infty$, then each fixed point $p\in [a,b]\setminus [[a,b]]$ of $T$ is extreme for $[a,b]$. Then, for the dynamical system generated by $T$, each positive semi-orbit starting in $[[a,b]]$ converges to a fixed point of $T$ in $[[a,b]]\cup \{ a,b \}$. The proof of this result combines ejective fixed-point theory with some geometric properties of maximal unordered subsets of $[[a,b]]$. Applications include large-time asymptotic behavior of competition between two species modeled by a time-periodic competitive system of two weakly coupled reaction-diffusion equations. The questions of extinction of one of the two species and unstable or stable coexistence of both species as well as the compressive case are discussed.

The purpose of this note is to present new results on linear transport equations with initial values in $W^{1,r}(\Omega)$ spaces, and coefficients in $W^{s+1,p}(\Omega)^N$, where $\Omega$ is a bounded open subset of $\mathbb{R}^N$ ($N \geq 2$) and $sp=N$. As an application, we also give refined regularity and uniqueness results for weak solutions of the two-dimensional density-dependent incompressible Navier-Stokes equations.

Global existence results for weak solutions of the so-called incompressible density-dependent Navier-Stokes equations are proven in the whole space $\mathbb{R}^N$ $(N \geq 2)$. In this note, the initial density is not required to be bounded from below by a positive constant and the viscosity may be a function of the density.