The following results are achieved under this grant.

(1) Study on the applications of “L-infinity Energy Method” to nonlinear partial differential equations:

   We applied L-infinity Energy Method to the system of parabolic equations which describes a diffusion-convection prey-predator model which takes into account of the hysteresis effects. Because of the strong nonlinearity of this system, we needed to improve some tools to establish a priori estimates for the L-infinity norm of solutions, by which we could ameliorate previous studies.

(2) Study on complex Ginzburg-Landau equations (CGLE):

  (i) For the non-dissipative system in bounded domains, we proved the existence and the uniqueness of time-local solutions for CGLE in H^1-space.

  (ii) We analyzed the finite-time blow-up of solutions of CGLE.

The previous studies dealt with the case where the energy of the initial data is negative in the whole domain.

We developed a new method to treat the case where the energy of the initial data is positive in general domains.

(3) Study on the mathematical analysis for the   mitochondrial  swelling   model:  

   We analyzed this model with Robin-type boundary conditions, which describes well the real situation of mitochondria in cells. We showed the well posedness of the system as well as the asymptotic behavior of solutions.