Long-time Dynamical Behavior Of Global Solutions For The Equation With Fading Memory

In this doctoral dissertation,applying the recent theoretical results about attractors and combining with some estimates of energy functional,we mainly consider the two classes of equations as follows:the non-classical diffusion equations with fading memory and the semilinear heat equations with fading memory.Long-time behavior of weak solution and strong solution is discussed and the existence of global attractors or uniform attractors respectively for either autonomous or non-autonomous case is gained.At first,we research the non-classical diffusion equations with fading memory u_t-â–³u_t-â–³u-integral from n=0 to∞k(s)â–³u(t-s)ds=f(u)+g(x) for the autonomous case.Because nonlinear term satisfies critical exponential growth condition and forcing term g only belongs to H^-1(Ω) or L²(Ω),there exist some virtuality difficulties in the process of proof.Conquering above difficulties through decomposition technique of semigroup and compactness transition theorem, furthermore,we prove the existence of global attractors in weak topological space and in strong topological space(see Theorem 3.2.9 and Theorem 3.3.8). After that,we discuss the non-classical diffusion equations with fading memory u_t-â–³u_t-â–³u-integral from n=0 to∞k(s)â–³u(t-s)ds= f(u)+g(x,t) for the non-autonomous case.When the nonlinear term satisfies critical exponential growth condition and the time-dependant forcing term is translation bounded(that is,only belongs to n_b²(R;L²(Ω)) or L_b²(R;H₀¹(Ω))) instead of translation compact,after decomposing the solution process we test and verify the asymptotic regularity of solutions.Based on the result we show the existence of compactly uniform attractors together their structure in both weak and strong topological spaces(see Theorem 4.2.15 and Theorem 4.3.5).In the end,we consider the semilinear heat equations with fading memory u_t-â–³u-integral from n=0 to∞k(s)â–³u(t-s)ds+f(u)=g(x).When the nonlinearity adheres to polynomial growth of arbitrary order,applying abstract semigroup theory,we make the priori estimate and obtain the existence and uniqueness of solutions. And then the asymptotic compactness of solution semigroup is gained by utilizing contract function theory.According to these results as indicated above, we show the existence of global attractors in both L²(Ω)×L_μ²(R⁺;H₀¹(Ω)) and H₀¹(Ω)×L_μ²(R⁺;D(A))(see Theorem 5.2.9 and Theorem 5.3.7).Since the equations we study satisfy the weaker assumptions as follows:the nonlinearity adheres to critical exponential growth or polynomial growth of arbitrary order;and for the autonomous case,the forcing term only belong the lower regularity space,or for the non-autonomous case,the forcing term is only translation bounded,these results we gain improve and extend some known results extremely.The principal tools in my paper are:abstract semigroup theory,compactness transition theorem and contract function.