Blow Up Of A Class Of Semilinear Parabolic Equation

Study the global existence and non-existence of solutions to Semilinear parabolic equation. Especially we prove the well posedness of non-local Semilinear parabolic equation in bounded domain. By using the estimates method in BV functional space and Gamma convergence technique we not only obtain the local existence and the uniqueness of solutions to the problem, but also prove the global existence and decay property of solutions to the problem. In particular, we give the sufficient conditions for blow up of solutions, which implies the relationship between blow up and decay of energy. Using the minimizer technology we can find the sharp bound of the energy of the system that we can control the kinetic energy to get the global existence and energy decay for the problem with small initial data. If the negative condition is satisfied, then the problem does not admit any global solution. By giving a sufficient condition of local nonexistence of solutions we can explain that the solutions do not exist beyond a point of time. Then we can use the estimates skills and the special differential inequality which can be satisfied, to get the finite time blow up finally.