Considered herein is a modified two-component periodic Camassa-Holmsystem with peakons. The local well-posedness and low regularity result of solutionsare established. The precise blow-up scenarios of strong solutions and several resultsof blow-up solutions with certain initial profiles are described in detail and the exactblow-up rate is also obtained.

We apply functional separation of variables within the approach of the group foliation method to the nonlinear wave equation with variable speed and external force: utt=A(x)(D(u)ux)x+B(x)Q(u), Ax≠0. A classification of these equations admitting functionally separable solutions is performed and the resulting solutions are obtained in explicit form in many cases.

This paper discusses a class of (n+1)-dimensional nonlinear diffusion equations with source term which arises in nonlinear shear flows of non-Newtonian fluids. It is shown that some radially symmetric equations admit certain types of conditional Lie B

In this paper, the potential symmetry method is developed to study systems of nonlinear diffusion equations. Potential variables of the systems are introduced through conservation laws; such conservation laws yield equivalent systemsauxiliary systems of PDEs with the given dependent and potential variables as new dependent variables. Lie point symmetries of the auxiliary systems which cannot be projected to the vector fields of the given dependent and independent variables yield potential symmetries of the systems. Classification for systems of nonlinear diffusion equations with two and three components is performed. Symmetry reductions associated with the potential symmetries are presented.

Consider the 1+1-dimensional quasi-linear diffusion equations with convection and source term ut=[um(ux)n]x+P(u)ux + Q(u), where P and Q are both smooth functions. We obtain conditions under which the equations admit the Lie Bǎcklund conditional symmetry with characteristic η=uxx+H(u)u2x+G(u)(ux)2-n+F(u)ux1-n and the Hamilton-Jacobi sign-invariant J=ut+A(u)uxn+1+B(u)ux+C(u) which preserves both signs, ≥0 and ≤0, on the solution manifold. As a result, the corresponding solutions associated with the symmetries are obtained explicitly, or they are reduced to solve two-dimensional dynamical systems.