The aim of this paper is to discuss some degenerate hyperbolic equation u_t + φ(u)_x = 0, where φ ∈ C¹(R \ {0}) ∩ C²(R \ {0}) is a nondecreasing function in R, where R = (-∞, +∞). Some entropy inequalities are obtained and can be applied to study the existence of local BV solutions of the above equation with local finite measures as initial conditions.

Liouville-type theorems for a class of semilinear elliptic systems are considered via the method of moving spheres.

In this paper, we obtain some linear estimates, trilinear estimates. Through these estimates, we prove the local wellposeness of modified Korteweg-de Vries equation in a quarter plane.

In this note we verify the key inequality Y_1(M) < Y_1(S^n) for the Yamabe constant Y_1(M) for manifolds M not conformal to the unit sphere, by using a solution to an associated equation as a test function.

This paper investigates the qualitative properties of solutions to certain quasilinear parabolic equations. Under appropriate conditions, we obtain that the solution either exists globally or blows up in finite time by making use of the energy method and subsolution techniques. We find out that the behavior of solution heavily depends on the sign and the growth rate of the nonlinear reaction term and the nonlinear flux through boundary at infinity.

In this paper the nonexistence for weak solutions of some degenerate evolution inequalities on groups of Heisenberg type is studied.

In this paper we consider the Cauchy problem for quasilinear hyperbolic systems with characteristics with constant multiplicity. Without restriction on characteristics with constant multiplicity (> 1), a blow-up result is obtained for the C¹ solution to the Cauchy problem under the assumptions where there is a simple genuinely nonlinear characteristic and the initial data possess certain weaker decaying properties.