The Research To A Class Of Nonlinear Parabolic Equations

The researches of this dizertation are two fold. In the first place, we prove the well-posedness and the decay estimates of the solution to the following equation:forλ= ?1 and p > 1.Well-posedness: For u0∈L~1(RN) L_∞(RN),λ∈R, there exists some a T > 0, such thatthe equation has a unique solution in the space X = C([0, T), L1(RN)) C([0, T), L_∞(RN)).Decay estimates: If Np > N +α+ max(m, -N), lim M(t) = M_∞> 0, then the anomalousdiFFusion term determines the large time asymptotics of the solutions; while the nonlineareFFects play a domination role, and lim M(t) = M_∞= 0 if 1 < p≤N +α+ m.In the second place, we study the blow-up criterion of the solution to the magneto-hydrodynamoc equationsthen the solution (u(x, t), b(x, t)), t∈(0, T) to the incompressible magneto-hydrodynamocequations can be continued to (0, T ), T > T.