Blowup solutions of Grushin's operator

In this note, we consider the blowup phenomenon of Grushin's operator. By using the knowledge of probability, we first get expression of heat kernel of Grushin's operator. Then by using the properties of heat kernel and suitable auxiliary function, we get that the solutions will blow up in finite time.Comment:

Global and Blow-Up Solutions for a Class of Nonlinear Parabolic Problems under Robin Boundary Condition

We discuss the global and blow-up solutions of the following nonlinear parabolic problems with a gradient term under Robin boundary conditions: (b(u))t=∇·(h(t)k(x)a(u)∇u)+f(x,u,|∇u|2,t), in D×(0,T), (∂u/∂n)+γu=0, on ∂D×(0,T), u(x,0)=u0(x)>0, in D¯, where D⊂RN  (N≥2) is a bounded domain with smooth boundary ∂D. Under some appropriate assumption on the functions f, h, k, b, and a and initial value u0, we obtain the sufficient conditions for the existence of a global solution, an upper estimate of the global solution, the sufficient conditions for the existence of a blow-up solution, an upper bound for “blow-up time,” and an upper estimate of “blow-up rate.” Our approach depends heavily on the maximum principles