The Existence Of Weak Solutions To Reaction Diffusion System Describing Reaction For Antibody And Virus

In this paper, we study the reaction diffusion systems which models thereaction between antibody and Virus. We state the main results and the proofs ofthe existence of solutions to four reaction – diffusion systems from ?u , ?? ( u )to ?pu in the diffusion terms of antibody equation. Furthermore, We prove theconvergence of the weak solutions when the reaction rate k → ∞.At first, we consider the reaction – diffusion problemu ( x ,0 )= 0 , v (x ,0)= v₀ ,u ( 0,t ) = φ( t), 0 < t < T, (1)where u is the concentration of antibody, v is the concentration of virus, k isthe reaction rate which is a positive constant, ? ( t ) is the prescribed positiveantibody concentration function.We assume that( H ) φ ( t ) is Ho|¨lder continuous, positive and non-decreasing for t >0.To look for solutions of reaction – diffusion systems, we introduce the newvariableIt transforms the reaction – diffusion system (1) to a new problem which is thefollowing reaction – diffusion equationz_t = z_xx-v₀(1-e^kz)z ( 0,t ) = Ψ ( t), 0 < t < T,z ( x ,0 )= 0 , 0 < x< ∞. (2)We can prove that problem (1) and problem (2) are equivalent.Giving another assumption to ? , ? satisfies the hypothesis( )H 0 ( [ )) ( ) ( )? ∈ C 2 +1 0, ∞ , ? ′ ≥ 0, ? 0 = 0, ?′0 =0.Thus the problem (2) has a unique solution( ) ( )z ∈ C 2 +α , H α2ST ∩C ∞ST , ( 0 < α< 1),where ST = {( x , t )0 < x < ∞ ,0 < t < T , T >0}.Under hypothesis ( )H 0, we can obtain some properties of solutions toproblem (2).If we assume that ? merely satisfies the hypothesis ( H ), we canapproximate ? by functions ?j which all satisfy ( )H 0 and are chosen suchthat( i ) ? j ( t )≤? ∞,( ii ) ? j ( t )→? ( j →∞).Let z j be the solution of reaction – diffusion System (2), which corresponds to?j. By compactness, there exists a convergent subsequence { }z m and a functionz Such that z m→z ( m → ∞).Plainly, z is the unique solution of problem (2). Consequently, we canconclude that problem (1) has a unique solution. Furthermore, using someauxiliary lemmas, we can find solution ( u ,v ) of problem (1) . We also get theexistence of the solution of problem (1).We can extend these results to the more general reaction – diffusion systemsut = u xx? kF ( u ,v), ( x , t )∈ sT,59vt = ? kF ( u ,v) , ( x , t )∈ sT,( )u 0,t = u0 , 0 < t < T,( )v x ,0 =v0 , x > 0. (3)Secondly, we consider the reaction – diffusion system in multi-dimensionaldomainsut = ?? ( u ) ? kF ( u ,v) , ( x , t )∈ QT ,vt = ? kF ( u ,v), ( x , t )∈ QT ,u = u0 , ( x , t ) ∈??× ( 0,T],u ( x ,0 )= 0, ( )v x ,0= v0, x ∈? . (4)Here QT = ?×( 0,T ],? is bounded in R nwith smooth boundary ?? , T ,u 0and v0 are positive constants. ? ( u ) = ∫ u0D ( s ) ds , in which D is the diffusioncoefficient of the medium, the parameter k stands for the reaction rate.We shall make the following assumptions about the functions D and F :( )H 1 ( [ ))D ∈ C 1 0,∞ and D ( s ) >0 wherever s >0. In addition, we requirethat D ( 0 ) + D ′( 0 )>0;( )H 2 ( ( ] ( ]) ( [ ] [ ])1F ∈ C 0, u 0 × 0, v0 ∩C α0, u 0 ×0,v0 for some α ∈ ( 0,1);( )H 3 F is monotonically non-decreasing with regard to the two variables;( )H 4 F ( 0, s ) = F ( s ,0 )=0 for [ ] [ ]s ∈ 0, v0 , s ∈0, u 0;and F ( u , v )>0 when( ) ( ] ( ]u , v ∈ 0, u 0 ×0,v0 .Since the reaction – diffusion system degenerates. Usually, It has noclassical solutions. Thus, we shall consider weak solutions in this paper.We are led to introduce a definition of a weak solutions, for u and v ,weintroduce the initial and boundary problem and the definition of week solutions ,u ( x,0 )=0,and we proved the existence of weak solutions and also the weak solutionsatisfies some properties. Using Bootstrap method, we can obtain the sequence ofthe weak solutions { }u m and { }vm , and then prove that they are monotone.We then let m →∞and obtain the existence of weak solutions to the reaction –diffusion system (4).At last, if we retain antibody equation and add the diffusion term to virusequation in the problem (4) ,we obtain a new reaction – diffusion systemut = ?? ( u ) ? kF ( u ,v), ( x , t )∈ QT ,vt = ?v ? kF ( u ,v), ( x , t )∈ QT ,u = u0 , v = 0 ,( x , t ) ∈??× ( 0,T] ,u ( x ,0 )= 0 , ( )v x ,0= v0 , x ∈? . (5)If we retain virus equation and use the equation to non – Newtonian fluid toreplace the antibody equation in problem (5) , we obtain another new reaction –diffusion system,( ) ( )ut = div ? u p?2 ? u ? kF u ,v, ( x , t )∈ QT ,vt = ?v ? kF ( u ,v) , ( x , t )∈ QT ,u = u0 , v = 0 , ( x , t ) ∈??× ( 0,T],u ( x ,0 )= 0 , ( )v x ,0= v0 , x ∈? . (6)Since the systems of reaction – diffusion equations (5) and (6) degenerate.We also consider their weak solutions. Similarly to the study of problem (4) , wecan proved the existence of weak solutions to the systems (5) and (6).Since the equations are introduced from the reaction between the antibodyand virus, then we consider the limit when k → ∞. Instancing the equation (6)and using some auxiliary lemmas, we proved that the sequence of weak solutionsconverges. And hence we verify that our model coincide with its background:In the process of the fast reaction, if the concentration of antibody is stillpositive, the concentration of virus must be zero. Otherwise if concentration ofvirus is still positive, then the concentration of antibody must be zero.