Decay Properties Of The Solutions To The Navier-Stokes Equations And The Damped Wave Equation

In this thesis, we will consider the long-time behavior of the mild solution to the Navier-Stokes equations and the decay properties of solutions for the damped wave equation with space-time dependent damping term.It is known that the Cauchy problem of the Navier-Stokes equations admits a unique local solution for the initial data in LN(RN). And recently, Lq(q∈[N,∞]) decay estimates of such solution, which is similar to a solution of the heat equation were established by Y. Giga and O. Sawada. In this thesis, we consider the difference between the mild solution to the Navier-Stokes equations and the solution to the heat equation. And we obtain the L2 decay rate of the difference, which is faster than that of the solution to the heat equation. Thus, we get the asymptotic profile of the mild solution to the Navier-Stokes equations. Moreover, we can construct higher approximation by a iteration in dimension 3.Besides, we consider the Cauchy problem for the damped wave equation with space-time dependent potential and absorbing semilinear term. Based on the local existence theorem, we obtain the global existence and the L2 decay rate of the solution by using the weighted energy method. From the decay rate we obtain, we find that the critical exponent isρc(N,α,β):= 1+2/(N -α), which coincides to the Fujita exponentρf(N):= 1+2/N when the potential is a constant. And when the potential is just a constant or a function dependent only on space or time, the decay rate we obtain coincides with the known results.