Supersonic-sonic patches are ubiquitous in regions of transonic flows and they boil down to a family of degenerate hyperbolic problems in regions surrounded by a streamline, a characteristic curve and a possible sonic curve. This paper establishes the global existence of solutions in a whole supersonic-sonic patch characterized by the two-dimensional full system of steady Euler equations and studies solution behaviors near sonic curves, depending on the proper choice of boundary data extracted from the airfoil problem and related contexts. New characteristic decompositions are developed for the full system and a delicate local partial hodograph transformation is introduced for the solution estimates. It is shown that the solution is uniformly $C^{1,\frac{1}{6}}$ continuous up to the sonic curve and the sonic curve is also $C^{1,\frac{1}{6}}$ continuous.

We are concerned with the stability of polytropic filtration equation with variable exponents, when the degeneracy occurs on the boundary. The existence and the local integrability of weak solutions are established. The stability of weak solutions independent of the boundary value condition is investigated.

This paper is to find new exact traveling wave solutions of the nonlinear Gerdjikov-Ivanov (for short, GI) equation with perturbation terms. Based on employing the bifurcation theory of planar dynamical systems, we found the exact solutions including periodic wave solution, kink wave solution, anti-kink wave solution and solitary wave solution (bright and dark). Moreover, the explicit expressions of the exact solutions in different parametric domains are given. Finally, we conclude our main results in a theorem at the end of the paper.

This paper mainly studies the explicit wave-breaking mechanism and dynamical behavior of solutions near the explicit self-similar singularity for the Camassa-Holm and Degasperis-Procesi equations, which can be regarded as a model for shallow water dynamics and arising from the approximation of the Hamiltonian for Euler's equation in the shallow water regime. We prove that the Camassa-Holm and Degasperis-Procesi equations admit stable explicit self-similar solutions. After that, the nonlinear instability of explicit self-similar solution for the Korteweg-de Vries equation is given.