Existence Of Uniformly L~? Bounded Weak Solutions To Some Degenerate Keller-segel Models

In recent years,when interdisciplinary researches become popular all over the world,more and more mathematicians focus themselves on models related to subjects,such as biology,chemistry and physics.Throughout this paper,we study an interesting biomath-ematics model called the Keller-Segel model,which is about the chemotaxis of bacteria.Keller-Segel model was firstly presented by Keller and Segel in 1970[1,2]to describe the chemotaxis of cellular slime molds.In this model,cells are attracted by a kind of chemical substance and also able to emit it.Our main purpose of this paper is to prove the existence of weak solutions to two different kinds of degenerate Keller-Segel models.The paper is organized as follows.In Chapter 1,we introduce the background information of the Keller-Segel model from 1970 until now.By presenting the construction process of the origin model,we hope readers can understand the model deeply and thorough if they want.We also point out some famous simplified models and elegant results to show the beauty of the Keller-Segel model,even to attract potential mathematicians to work on it.After that,we illustrate the origin of the inspiration,difficulties to be overcome and conclusions obtained in the paper.Some open problems are given in this part.In Chapter 2,we study the following degenerate parabolic-parabolic Keller-Segel model in d>3:where the diffusion exponent m is taken to be 0<m<2-2/d.In the model,u(x,t)repre-sents the cell density and v(x,t)represents the concentration of the chemical substance.Since cells are attracted by the chemical substance and also able to emit it,without loss of generality,we suppose v(x,0)= 0 which is reasonable with the meaning that there is no chemical substance at the beginning,and then it is generated by cells.In order to prove the existence of global weak solutions,we should obtain a priori estimates of weak solutions at first.For degenerate parabolic-elliptic Keller-Segel models which have been widely studied,the Hardy-Littlewood-Sobolev inequality with sharp constant is the key in this process:While in the degenerate parabolic-parabolic Keller-Segel model,the HLS inequality does not work since v(x,t)could not be represented by the fundamental solution.Thus,we use Semi-group Theory instead of HLS inequality to obtain a priori estimates.The following definition and estimates are standard.Consider the following Cauchy problem:Definition 0.0.1.Let T>0,p ? 1,h0 ?Lp(Rd)and f E L2(0,T;L2(Rd)).The function h(x,t)? C([0,T];L2(Rd))given by is the unique mild solution of problem(9)on[0,T].The heat semigroup operator et? is defined by(et?f)(x,t):=G(x,t)*f(x,t),where G(x,t)is the heat kernel by G(x,t)=1/((4?t)d/2)e-(|x|2)/(4t).It is not hard to check that the mild solution defined above is also a weak solution.Now we introduce the famous Maximal LP-regularity Theorem which is the key in the process of a p,riori estimates.Lemma 0.0.1.Let 1<p<+? and T>0.Then for each f? Lp(0,T;LP(Rd)),problem(9)has a unique solution h(x,t)with h0(x)= 0 in the LP(0,T;Lp(Rd))sense.Moreover,there exists a positive constant Cp such thatUntil now,with the maximal LP-regularity and standard estimates,we obtain the a priori estimates for weak solutions of(8):Theorem 0.01.Let d?3,0<m<2-2/d and p=(d(2-m))/2.Cp is the posi-tive constant in(10).Under the assumption that u0?L+1?Lp(Rd)and ?=Cd,m2-m-?u0?Lp(Rd)2+m>0,where Cd,m2-m=4mp/(Sd-1(m+p-1)2Cp)is a universal constant,let(u,v)be anon-negative weak solution of(8).Then u ?L?(R+iLp(Rd)),u?Lp+1(R+;Lp+1(Rd))and ?u(m+p-1)/2?L2(R+;L2(Rd)).It is known that L1-norm and L?-norm estimates of weak solutions are two important properties.The mass conservation is obtained as a by product in the process of a priri estimates.Then we use Bootstrap iterative method to prove that weak solutions are uniformly L? bounded.Theorem 0.0.2.Let d?3,0<m<2-2/d and p=(d(2-m))/2.If u0?L+1(Rd)?L?(Rd)and ?=Cd,m2-m-?u0?Lp(Rd)2-m>0,where Cd,m2-m=4mp/(Sd-1(m+p-1)2Cp)is a universal constant,suppose(u,v)be a non-negative weak solution of(8).Then for anny t>0,?u(·,t)?L?(Rd)?C(m,d,K0),where K0=max(?).With all the a priori estimates of weak solutions obtained,we could prove our main theorem in Chapter 2 about the existence of global weak solutions of(8)by constructing a corresponding regularized problem:for ?>0,where d?3,0<m<2-2/d.With some appropriate assumptions of the initial datau0?(x),we obtain that the regularized problem has a classic solution which satisfies all a priori estimates in Theorem 0.0.1.The difficulty we encounter in the proof is that the Aubin-Lions Lemma does not work in our situation when proving the strong convergence,since we only have the uniform boundedness of ??ut(m+p-1)/2?L2(R+;L2(Rd))instead of the uniform bounded estimate of ?u?.Thus,we introduce the Aubin-Lions-Dubinskii Lemma[3]:Lemma 0.0.2.Let B,Y be Banach spaces and M+ be a seminormed nonnegative cone in B with M+ ?Y ?(?).Let 1 ? p ? ?· We assume that(?)M+?Bcompactly.(?)For all(?n)(?)B,?n?? in B,?n?0 in Y as n?? imply tha ?=0.(?)U(?)Lp(0,T;M+?Y)is bounded in Lp(0,T;M+).(?)||u(t+h)-u(t)||Lp(0,T-h;Y)?0 as h?0,uniformly in u ? U.Then U is relatively compact in LP(0,T;B)(and in C0([0,T];B)if p =?).In order to apply the Aubin-Lions-Dubinskii Lemma,we choose B =Lp+1(?)and construct which is a Seminormed non-negative cone in Lp+1(?)that satisfies the definition:Definition 0.0.2.Let B be 'a Ba,nach space,M+ C B satisfies(1)Cu ? M+,for all u ?M+ and C?0,(2)there exists a function[·];M+ ?[0,?)such that[u]=0 if an,d only if u = 0,(3)[Cu]=C[u],for all C?0,then M+ is a Seminormed non-negative cone in B.By using Aubin-Lions-Dubinskii Lemma,we prove the existence of global weak so-lutions step by step.Furthermore,when 1<m<2-2/d,the weak solution is also anentropy solution which satisfies entropy-dissipation inequality.We have listed all crucial ideas in the proof of the main existence theorem in Chapter 2,now we give the complete version of the existence theorem below:Theorem 0.0.3.Let d?3,0<m<2-2/d and p=(d(2-m)/2).Assume u0?L1+?Lp(Rd)and ?=Cd,m2-m-?u0?Lp(Rd)2-m>0,where Cd,m2-m=4mp/(Sd-1(m+p-1)2Cp))is a universal constant.Then there exists a non-negative global weak solution(u,v)of(8),such that all the a priori estimates in Therem 0.01 hold true,Furthermore,for 1<m<2-2/d,if the initial date satisfies ?Rd|x|2u0(x)dx<?,and ?u0?Ld/2(Rd)?C,then(?)the second moments dx are bounded for any 0 ?t<?,(?)the free energy of(8)is which is non-increasing in time,(?)with an extra assumption that u0 ? Lm(Rd)when 2d/d+2<m<2-2/d,for all1<m<2-2/d the weak solution of(8)also satisfies energy inequalityAt the end of the Chapter 2,we prove the local existence of weak solutions and a blow-up criterion.The finite time blow-up of degenerate parabolic-parabolic Keller-Segelmodel when 0<m<2-2/d is still an open problem.In Chapter 3,we propose and study the p-Laplacian Keller-Segel model in d? 3 where p>1.It is a natural extension from degenerate parabolic-elliptic Keller-Segel model since the porous medium equation and the p-Laplacian equation are all called nonlinear diffu-sion equations.Work in these two models has frequent overlaps both in phenomena to be described,results to be proved and techniques to be used.In the p-Laplacian Keller-Segel model,we find out a critical exponent p which plays a similar role as m = 2-2/d in(8).When p =3d/d+1,if(u,v)is a solution of(12),constructing the following mass invariant scaling for u and a corresponding scaling for v then(u?,v?)is also a solution for(12)and hence p = 3d/d+1 is referred to the critical exponent.For the general exponent p,(u?,v?)satisfies the following equation which is called the supercritical case,the aggregation dominates the dif-fusion for high density which leads to the finite-time blow-up,and the diffusion dominates the aggregation for low density which leads to the infinite-time spreading.which is called the subcritical case,the aggregation dominates the diffusion for low density which prevents spreading,while the diffusion dominates the aggregation for high density which prevents blow-up.In Chapter 3,our main goal is to prove the existence of global weak solutions of(12)with big initial data in the supercritical case.In order to prove the existence theorem,we have a priori estimates of weak solutions at first:Theorem 0.04.Let d?3,1<p<(3d)/(d+1)and q=(d(3-p)/p.Under the assumption that u0 ?L+1(Rd)and A(d,p)=Cp,d3-p-?u0?Lq3-p>0,where Cp,d=(?)is a universal constant,let(u,v)be a non-negative weak solution of(12).Then u ?In the p-Laplacian Keller-Segel equation,we do not have the mass conservation result of u like(8)do.This is an open problem.Using the Bootstrap iterative method,we also prove that weak solutions of(12)are uniformly L? bounded.The main ideas in the proof are similar with ones in Theorem 0.0.2,but details are not quite the same.Theorem 0.0.5.Let d?3,1<p<(3d)/(d+1)and q=(d(3-p)/p).If u0?L+(Rd)?L?(Rd)and A(d,p)=Cp,d3-p-?u0?Lq(3-p)>0,where Cp,d=[(qpp)/Kp(d,p)(q-2+p)p]1/3-p is auniversal constant,let(u,v)be a non-negative weak solution of(12).Then for any t>0After obtaining all a priori estimates of weak solutions,we construct a corresponding regularized problem of(12)to prove the main existence theorem in Chapter 3:for ?>0 where d?3,1<p<(3d)/(d+1)and J?(x)=1/(?d)J(x/?),J(x)=1/(?(d))(1+|x|2)-((d+2)/2)satisfying?Rd J?(x)=1.A simple computation shouws that v? can be expressed by where a(d)is the volume of the d-dimensional unit ball.With the appropriate.assumptions of initial data u0?(x),we prove that all a priori estimates in the Theorem 0.0:4 hold true for the classic solution of the regularized problem.Then combining the strong convergence which obtained from Aubin-Lions Lemma and weak convergence from uniformly bounded estimates,we finish the proof of our main theorem in Chapter 3:Theorem 0.0.6.Let d ?3,1<p<3d/d+1 and q = d(3-p)/p.If u0?L+1(Rd)?L?(Rd)and A(d,p)=Cp,d3-p-?u0?Lq3-p>0,where Cp,d=(?)is a universal constant.Then there exists a non-negative global weak solution(u,v)of(12),such that all a priori estimates in Theorem 0.0.4 and the uniform L? estimate in Theorem 0.0.5 hold true.The tricky part in the proof is to use the monotone operator to obtain the limit of the nonlinear term.The following lemma is the useful property of the monotone operator:Lemma 0.0.3.For any ?,?'?Rd,there existswhere C1 and C2 are two positive constants only depending on p.For p-Laplacian Keller-Segel model with 1<p<(3d)/(d+1),the finite time blow-up result of the weak solution is still an open problem.