Complex Asymptotic Behavior Of Solutions To Fractional Porous Media Equation

In this paper,we consider the complicated asymptotic behavior of the solution of the Cauchy problem to the fractional porous medium equation(?),where ▽α-1 stands for ▽(-Δ)α/2-1,α∈(0,2).The fractional porous medium equation is a kind of fractional differential equation combining nonlocal operator and porous medium,which can well explain the natural phenomena and change laws of some special materials with heredity and memory,and has many applications in mathematics,physics and other fields However,existing research on the fractional porous medium equation mainly focuses on the basic properties of the existence and uniqueness,regularity,and simple asymptotic behavior of solutions.However,there is little attention paid to the complicated asymptotic behavior of solutions.Moreover,in the field of research on the complicated asymptotic behavior of solutions to differential equations,the relevant research results of the fractional differential equations are far less than those of integer order differential equations.Therefore,it is necessary to explore the complicated asymptotic behavior of solutions to the fractional porous medium equation.The main purpose of this article is to reveal the qualitative properties of solutions to the fractional porous medium equation,and to characterize the complicated phenomenon that the solutions of such fractional order differential equations exhibit over time.The specific research content is as follows:The rescaled solution and aggregation point set obtained by scaling the fractional porous medium equation are defined.By using the Riesz potential to represent nonlocal operators,the commutative relationship between scale transformation operators and semigroup operators was proved.When the initial value belongs to L∞(RN)space or L1(RN)space,the propagation velocity estimation of the solution support of the Cauchy problem of the fractional porous medium equation is established.The concrete form of the initial value u0 is constructed in the subset C0+(RN)of the continuous function space.Combining the superposition property,decay estimates and regularity of the solution,it is obtained that under the condition of 0<μ<2N/(N+α),β>(2-μ)/2α,the ω limit set of the scale solution tnμ/2u(tnβχ,tn)contains more than one element,which proves that the solution of the Cauchy problem of the fractional porous medium may have complicated asymptotic behavior in C0+(RN).By using the method of constructing initial values,the structure of the aggregation point set of the scale solution is obtained.That is,for any countable subset F of(0,2Nα/(N+α)(2-μ)),when the scale transformation parameters still satisfies μ/2β∈F,the ω limit set is equivalent to C0+(RN),further revealing the complexity of asymptotic convergence of solutions to such equations.In conclusion,this paper analyzes the complicated asymptotic behavior of the solution of the Cauchy problem of the fractional porous medium equation.On the one hand,it improves the theory of the fractional porous medium equation and provides an effective reference for the practical application of this kind of equation.On the other hand,it also promotes the development of the theoretical research on the fractional differential equation.