The Initial Value Problem And The Initial Boundary Value Problem For A Class Of Nonlinear Parabolic Equations Of Higher Order

This paper consists of three chapters. The first chapter is the introdution. In the second chapter,we will study the existence and uniqueness of the local solution to the initial value problem for a class of nonlinear parabolic equations of higher order,we will prove the existence and uniqueness of the global solution to the problem by the extension therem of the solution,and we will discuss the decay property of the solution.In the third chapter,we will study the existence and uniqueness of the local solution to the initial value boundary problem for this class of nonlinear parabolic equations of higher order,we will prove the existence and uniqueness of the global solution to the problem by the extension therem of the solution,and we also will study the decay property of the solution.In the second chapter,we study the existence and uniqueness of the local solution and the global solution of the following initial value problem for a class, of nonlinear parabolic equations of higher orderwhereα> 0,β> 0,γ> 0 are constants,f(s),h(s), G(s) and g(s) are given nonlinear functions,u₀(x) is given initial value function.The main results are the following:Theorem 1 Assume that u₀∈H^s(R)(s > 3/2), g, f,h,G∈C^k(R) and k = [s] + 1,then the initial value problem(1), (2)has a unique local solution u(x,t)∈C([0,T₀); H^s(R))∩ where [0, T₀) is a maximal time interval. Moreover, ifthen T₀ =∞.Theorem 2 Assume thatwhere u(x,t) and T₀ are mentioned in Theorem 1.ThenTheorem 3 Assume that u₀∈H^s(R)(s≥2), g, f, h, G∈C^k(R) and k = [s] + 1, h(0) = 0 and ,there is constantγ₀> 0,such that .Then the problem (1), (2)has a unique global solution u(x,t) .Theorem 4 Assume that(i) h∈C¹(R), h(0) = 0 and ; g∈C¹(R) and g' (ξ)≥0.(ii) G∈C¹(R), G(0) = 0 and there is constantγ₀> 0,such that G'(ξ)≤-γ₀,. Then the global solution of the problem (1),(2) has the decay propertyIn the third chapter,we discuss the initial value boundary problem The main results are the following:Theorem 5 Assume that u₀∈H³(Ω), f∈C²(R), g∈C³(R), h∈C²(R) and G∈C¹(R),then the problem (6)-(8) has a unique local solution ,where [0, T₀) is a maximal time interval. Moreover, ifthen T₀ =∞.Theorem 6 Assume that and G∈C¹(R) and G(0) = 0,there are constants C₁,C < 0,A, B, D, such that G'(s)≤A; h'(s)≥B; g'(s)≥D and C≤f'(s)≤C₁, (?)s∈R,then the problem (6) - (8) has a unique global solution .Theorem 7 Assume that and G∈C¹(R) and G(0) = 0,there are constants C₁,C < 0,A, B, D, such that G'(s)≤A, h'(s)≥B, g'(s)≥D and C≤f'(s)≤C₁,(?)s∈R , where 2A+1 <0,(C²)/2 - B - D -β< 0,then the global solution of the problem (6)-(8) has the decay propertywhere...