In this paper, we investigate the existence of global weak solutions to an integrable two-component Camassa–Holm shallow-water system, provided the initial data u0(x) and ρ0(x) have end states u± and ρ±, respectively. By perturbing the Cauchy problem of the system around rarefaction waves of the well-known Burgers equation, we obtain a global weak solution for the system under the assumptions u− ≤ u+ and ρ− ≤ ρ+.

It is well known that boundary value problems for hyperbolic equations are in general “not well posed” problems. This paper is concerned with the uniqueness of solutions to boundary value problems for the hyperbolic equation uxx − Qu = utt. Here Q is a function of the variable x alone, and satisfies the following conditions:

(a) Q:[0, ∞) → ℝ;

(b) Q is Lebesgue integrable on any compact subinterval of [0, ∞);

(c) Q(x)→ ∞ as x → ∞.

We investigate the ‘clumping versus local finiteness' behavior in the infinite backward tree for a class of branching particle systems in ℝd with symmetric stable migration and critical ‘genuine multitype' branching. Under mild assumptions on the branching we establish, by analysing certain ergodic properties of the individual ancestral process, a critical dimension d c such that the (measure-valued) tree-top is almost surely locally finite if and only if d > d c. This result is used to obtain L 1-norm asymptotics of a corresponding class of systems of non-linear partial differential equations.

For a class of formally hypoelliptic differential operators in divergence form we prove a generalized Gårding inequality. Using this inequality and further properties of the sesquilinear form generated by the differential operator a generalized homogeneous Dirichlet problem is treated in a suitable Hilbert space. In particular Fredholm's alternative theorem is proved to be valid.