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This article concerns the null controllability of a control system governed by coupled degenerate parabolic equations with lower order terms. For these equations, the convection terms cannot be controlled by the diffusion terms. We establish a Carleman estimate and an observability inequality by using Carleman estimates for single degenerate parabolic equations with lower order term and some energy estimates. Then we prove that the system with two controls is null controllable. Finally, we show the null controllability of the system with one control, by constructing suitable controls.

This article shows Carleman estimate and null controllability of a cascade control system governed by the semilinear degenerate parabolic equations with the general convection terms. The semilinear parabolic equations are weakly degenerate on the boundary and the convection terms cannot be controlled by the diffusion terms. We establish the Carleman estimate and the observability inequality for the linear conjugate system, and prove that the control system is null controllable.

This article concerns the asymptotic behavior of solutions to the Cauchy problem of a degenerate quasilinear parabolic equations with a gradient term. A blow-up theorem of Fujita type is established and the critical Fujita exponent is formulated by the spacial dimension and the behavior of the coefficient of the gradient term at infinity.

This article concerns the quenching phenomenon of the solution to the Dirichlet problem of a semilinear diffusion equation with convection and reaction. It is shown that there exists a critical length for the spatial interval in the sense that the solution exists globally in time if the length of the spatial interval is less than this number while the solution quenches if the length is greater than this number. For the solution quenching at a finite time, we study the location of the quenching points and the blowing up of the derivative of the solution with respect to the time.

This article concerns the quenching phenomenon of the solution to the Dirichlet problem of a singular and degenerate quasilinear diffusion equation. It is shown that there exists a critical length for the special domain in the sense that the solution exists globally in time if the length of the special domain is less than this number while the solution quenches if the length is greater than this number. Furthermore, we also study the quenching properties for the quenching solution, including the location of the quenching points and the blowing up of the derivative of the solution with respect to the time at the quenching time.