EXPONENTIAL ATTRACTORS FOR COMPETING SPECIES MODEL WITH CROSS-DIFFUSIONS

. This paper is concerned with the competing species model presented by Shigesada-Kawasaki-Teramoto in 1979. Under a suitable condition on self-diﬀusions and cross-diﬀusions, we construct a dynamical system determined from the model. Furthermore, under the same condition we construct exponential attractors of the dynamical system.

in a two-dimensional bounded C 3 domain Ω ⊂ R 2 .Here, a > 0, b > 0, α ij ≥ 0, c > 0, d > 0 and γ ij > 0 are given constants.In 1979, this system was introduced by Shigesada-Kawasaki-Teramoto [18] to describe the segregation process of a biological system consisting of two competing species, say A and B, in Ω by the cross-diffusions.The unknown functions u = u(x, t) and v = v(x, t) denote the densities of species A and B in Ω at time t ≥ 0, respectively.They are subjected to homogeneous Neumann boundary conditions on ∂Ω.The terms α ij ∆uv (i = j) denote the cross-diffusions between A and B. On the other hand, the terms α 11 ∆u 2 and α 22 ∆v 2 denote the self-diffusions of A and B, respectively.The competitions of species are described by the two kinetic functions (c − γ 11 u − γ 12 v)u and (d − γ 21 u − γ 22 v)v.The initial functions u 0 and v 0 are given in such a way that u 0 v 0 is in the product space where ε denotes an arbitrarily fixed exponent in such a way that 0 < ε < 1 2 .Problem (1) is handled in the product space of L 2 (Ω), i.e., This system has in fact attracted interest of many mathematicians for these thirty years.For the two-dimensional problem, the global existence for all initial values in K is known so far under the following condition of α ij : 0 ≤ α 12 α 21 ≤ 64 α 11 α 22 . (4) Note that, if α 12 α 21 = 0, i.e., one of the cross-diffusions does not exists, then it is allowed that α 11 = α 22 = 0.A global existence result was first obtained by the author [25] (cf.also [26]) in the case when 0 < α 21 < 8α 11 and 0 < α 12 < 8α 22 ; afterward, this result was extended by Ichikawa-Yamada [8] to the case when 0 < α 12 α 21 < 64α 11 α 22 .For the critical case α 12 α 21 = 0, the global existence result was first obtained by Masuda-Mimura (cf.[13]) for the one-dimensional problem.For the two-dimensional case, this was shown by the author [27]; afterward, the similar result was shown by Lou-Ni-Wu [12] but in a framework of the L p (2 < p < ∞) theory.For the other critical case 0 < α 12 α 21 = 64α 11 α 22 , this will be seen in the present paper.
It is then very natural to ask whether Condition ( 4) is necessary for the global existence of solutions for all U 0 ∈ K or not.In a particular case when a = b, α 12 α 21 > 0, α 11 = α 22 = 0, Kim [9] proved the global existence for the one-dimensional problem.Such a result can be extended to the two-dimensional one, too (cf.[25,Remark 4.6]).But, for the moment, it is very difficult to give a satisfactory answer to the question; for example, no blowup results for (1) are known.
For the N -dimensional problem (N ≥ 3), Deuring [5] first considered the global existence in the case when α 12 and α 21 are sufficiently small.Afterward, Pozio-Tesei [16] got rid of such smallness (under the Dirichlet boundary conditions) but assuming a higher order of decaying in the growth function of u or v. Yamada [24] also studied the problem in the same spirit.Wiegner [22] applied the Amann's theory [1] on abstract parabolic equations to the N -dimensional problem of (1).More recently, Choi-Lui-Yamada [3,4] tried to extend the result [12] to the Ndimension case; but still they need smallness of α 12 > 0 (α 21 = 0) or positivity of α 11 > 0.
In the meantime, little is known for the asymptotic behavior of solutions.Redlinger [17] constructed the global attractor for the one-dimensional problem containing growth functions at a higher order of decaying.Shim [19,20,21] established uniformly bounded estimates and convergence of solutions in some suitable cases.
A number of papers on the stationary problem for (1) have already been published.We will here quote only some of them such as [10,11,13,14,23].For the full references we refer the reader to References therein.
This paper is then concerned with constructing a dynamical system determined from (1) in the two-dimensional case and constructing exponential attractors for the dynamical system.We also consider the case when the cross-diffusion coefficients and the self-diffusion coefficients satisfy Condition (4).The notion of exponential attractors has been introduced in 1994 by Eden-Foias-Nicolaenko-Temam [29].The exponential attractor is a compact, positively invariant set of finite fractal dimension containing the global attractor and attracting every trajectory at an exponential rate.It is also known that the exponential attractor enjoys stronger robustness with respect to the global attractor, see [29] and also, e.g., [6,7].
As shown in [25,27], local existence of solutions for (1) is obtained by directly applying the general results concerning abstract parabolic evolution equations.It is however necessary to verify that a realisation of the matrix differential operator in the space X is a sectorial operator with angle < π 2 .This verification is not immediate.Especially in the case of α 12 α 21 > 0, namely, the matrix is a full matrix, we need to use some techniques.In showing the local existence, Condition ( 4) is not at all necessary.For any U 0 ∈ K, a unique local solution to (1) can be constructed.
For constructing global solutions, we have to build up norm estimates (for the local solutions) which ensure that the H 1+ε norms of solutions never blow up in finite time.For constructing the global attractor (a fortiori, the exponential attractor), we have to build up stronger norm estimates of solutions which show that the H 1+ε norms of solutions having large norms of initial data decrease asymptotically and become smaller than a universal constant, say C > 0, which is independent of solutions as t → ∞.As a matter of fact, building up such a priori estimates will occupy the main part of the present paper.For this aim we need more careful calculations than before ( [25,27]) and need various techniques.The critical case α 12 α 21 = 0 of (4) may be more delicate than the other favorable case.We will employ analogous techniques utilized in [15] for the chemotaxis-growth model.
In constructing exponential attractors, we know two kinds of sufficient conditions concerning the nonlinear semigroup of the dynamical system under consideration.The first one is the squeezing property which has been introduced by Eden-Foias-Nicolaenko-Temam in the mentioned book [29].The second one is the compact smoothing property, see (83), introduced by Efendiev-Miranville-Zelik [6].In the sense of logic, these two properties are mutually equivalent when the universal space is a Hilbert space.But, in the viewpoint of applications, the squeezing property fits more to the semilinear diffusion equations than the quasilinear diffusion equations.So, to the present system, we will apply the compact smoothing property by verifying Condition (83).General procedure for verifying (83) was present in the paper Aida-Efendiev-Yagi [2] in which we utilized representing formulas of quasilinear abstract parabolic evolution equations provided by the semigroup methods.These methods are reviewed in Section 6 of this paper.The semigroup methods are known as powerful tools for solving parabolic equations and systems.We will just follow the general procedure to verify the compact smoothing property of our nonlinear semigroup.
The C 3 regularity of the boundary ∂Ω is needed only in the proof of the a priori estimates for the critical case α 12 α 21 = 0 of (4).All other results are valid under C 2 regularity of ∂Ω or even under convexity of Ω.The C 3 regularity of ∂Ω ensures the shift property that ∆u ∈ H 1 (Ω) with ∂u ∂n = 0 implies u ∈ H 3 (Ω).Such a shift property is used in Step 5 of the proof of Proposition 2.

Local solutions.
As a matter of fact, we already know (see [25,Section 3]) that the theory of quasilinear abstract parabolic evolution equations is available for constructing a unique local solution to (1) for any pair of u 0 and v 0 from K. Some results of the theory are reviewed in Subsection 6.1.
Fix an exponent ε ′ so that 0 < ε ′ < ε (remember that ε was already taken in (2)) and let For any 0 < R < ∞, let K R = {U ; U Z < R} be an open ball of Z.For each U ∈ K R , let A(U ) denote a closed linear operator given by [25, (3.4)].Let F be a nonlinear operator from K R into X given by [25, (3.7)].
Let us take any initial value U 0 ∈ K and take an R sufficiently large so that U 0 ∈ K R .Then (1) can be written as the Cauchy problem of the form in the space X given by (3).Then, A(U ), U ∈ K R , are seen to be sectorial operators of X with angle < π 2 fulfilling (68) announced in Section 6.Their domains are given by where H 2 N (Ω) = {u ∈ H 2 (Ω); ∂u ∂n = 0 on ∂Ω}.Moreover, according to [25, Proposition 3.2], we see for any 0 ≤ θ < 3  4 that Similarly, the operator-valued function A(•) fulfills the Lipschitz condition of form (69) with Y = X, namely, with α = 0.The nonlinear operator F also fulfils (70) with Y = X.Finally, on account of (7), the initial value U 0 fulfils the compatibility condition given by (71) with γ = 1+ε 2 (note that ε < 1 2 ).Therefore, Corollary 1 in Section 6 provides existence and uniqueness of a local solution to (6).
Furthermore, the truncation method deduces nonnegativity of the local solution.In this way, as stated in [25,Theorem 3.5], for any U 0 ∈ K, (6) and hence (1) possesses a unique nonnegative local solution U in the function space: where T U0 > 0 is determined by the norm U 0 H 1+ε and the constant C U0 > 0 is also determined by U 0 H 1+ε .In addition, as shown in [25,Theorem 3.6], we can utilize the maximal regularity of abstract parabolic evolution equations to verify the following regularity where H 1 (Ω) * denotes the dual space of H 1 (Ω).
3. A priori estimates.We shall establish a priori estimates for local solutions to (1).Let U 0 ∈ K and let U denote any nonnegative local solution to (1) in the function space: (10) where [0, T U ] denotes the interval on which U is defined.As shown in the preceding section, such a local solution exists at least on some interval [0, T U0 ].
In this section, Assumption (4) will be used.But the techniques of proof are quite different depending on the cases when α 12 α 21 > 0 and when α 12 α 21 = 0.When 0 < α 12 α 21 ≤ 64α 11 α 22 , ( 4) is seen to be equivalent to (12).Furthermore, (12) implies nonnegativity of a quadratic function for the variables p and q (u and v being nonnegative parameters) given by (13).Such nonnegativity of the quadratic function will be used several times in the a priori estimates below.In the meantime, when α 12 α 21 = 0, it is allowed that α 11 = α 22 = 0.So the estimate of form ( 13) is no longer valid in general.But if α 21 = 0 (resp.α 12 = 0), we can derive an L ∞ -norm estimate of the solution v(t) (resp.u(t)) in a direct way, and we can use this estimate for deriving other norm estimates concerning partial derivatives of the solutions u(t) and v(t).
For simplicity, we shall use the following quadratic functions 3.1.Case when α 12 α 21 > 0. Let us begin with noticing some scaling property.Let λ > 0 and µ > 0 be two parameters and multiply the equations for u and v by λ and µ, respectively.Then we obtain an equivalent problem to (1): where u λ (x, t) = λu(x, t) and v µ (x, t) = µv(x, t).It is clear that, if 0 < α 12 α 21 ≤ 64α 11 α 22 , then the new self-diffusion constants and cross-diffusion constants given in (11) satisfy again the same condition.Under (4) with α 12 α 21 > 0, if we choose parameters λ and µ so that the relation This means that the self-diffusion constants and cross-diffusion constants appearing in (11) Since (12) clearly implies (4), ( 12) is a stronger assumption than (4).But, since (11) is completely equivalent to (1) as the initial-boundary value problem, any a priori estimates which hold for all local solutions to (11) hold equally for all local solutions to (1).We are thus allowed to assume Condition (12) instead of 0 < α 12 α 21 ≤ 64α 11 α 22 in establishing our a priori estimates for the local solutions of (1).
Proposition 1.Let (12) (or, as mentioned above, 0 < α 12 α 21 ≤ 64α 11 α 22 ) be satisfied.There exists a continuous increasing function p(•) such that, for any local solution U to (1) lying in (10) Proof.In the proof, a unified notation C will be used to denote various constants which are determined from the initial constants a, b, c, d, α ij (1 ≤ i, j ≤ 2) and γ ij (1 ≤ i, j ≤ 2) and the domain Ω alone in a specific way.So, C may change from occurrence to occurrence.When a constant C depends on a particular parameter, say ζ, we shall denote it by C ζ .Similarly, a unified notation p(•) will be used to denote various continuous increasing functions which may change from occurrence to occurrence.
Step 1.Consider the inner product of the two evolution equations in (1) and It then follows that To estimate the integral in the right hand side we notice the inequality Then, 11 |Ω|/54.As a similar estimate holds for the integral Ω v 2 dx also, we obtain that 1 2 Here we notice that ( 12) is a necessary and sufficient condition in order that the inequality 4(2α Therefore we conclude that the fourth integral in the left hand side is nonnegative.Hence, 1 2 In particular, where C is given precisely by Integrating ( 14) on (0, t), we as well conclude that Step 2. We next consider the inner product of the two evolution equations of (1) and d dt From the equation for u, A similar energy equality is valid for Q, too.From these two equalities it follows that 1 2 We can use (13) again to observe nonnegativity of the third integral in the left hand side.After obvious calculations, 1 2 Using the Gagliardo-Nirenberg inequality (cf.[25, (1.3)]), we notice from (15) that we observe that all the L ∞ -norms of the components of are estimated by a constant C (remember α 11 > 0 and α 22 > 0).Hence we can verify that Consequently, Of course a similar estimate holds for v(t) L6 , too.Therefore, Let us next consider the inner product of the two equations in (1) and with any number ζ > 0. So, for any parameter ξ > 0, it is possible to observe that ξ Furthermore, following the same arguments as for (18), it is shown that C ξ > 0 being another constant depending on the parameter ξ.Hence, Combining this estimate with (19), we obtain the following estimate 1 2 Regarding this as a differential inequality on Ω (|∇P | 2 + |∇Q| 2 )dx and solving it, we have Furthermore, we introduce the corresponding version of ( 16) obtained by integrating ( 14) on (s, t).In fact, thanks to (15), Therefore, Fix now the parameter ξ as ξ = {p( U 0 L2 ) + 1}/2.Then, thanks to (15) again, we conclude that Moreover, in view of (17), and At the same time, integrating (20) on (0, t) in view of ( 21) and ( 22), we conclude that In view of these estimates, we may introduce a notation Step 3. We now use the equation in H 1 (Ω) * satisfied by the derivative u t = ∂u ∂t .Consider the duality product of this equation and A similar energy equality holds for v t , too.We combine these.After some calculations, As before, (13) shows that the third integral in the left hand side is nonnegative.Therefore, By the Gagliardo-Nirenberg inequality, we notice that In view of Lemma 3.1 announced below and ( 22), we have From (21), with any ζ > 0. Handling in a similar way for the other integrals, we verify the estimate ).By the similar procedure (actually it may be easier), we verify also the estimate ).

EXPONENTIAL ATTRACTORS COMPETING MODEL 1101
Thus it has been shown that the derivatives u t and v t satisfy 1 2 ).Meanwhile, on account of ( 19), ( 21) and (22), observe that the following differential inequality holds, where δ > 0 is a positive exponent given by δ = min{a, b} and η > 0 is a parameter.
The previous two inequalities then yield the following one Solving this, we have Use the corresponding version of ( 23) obtained by integrating (20) on (s, t).In fact, thanks to (22), Hence, This means that, if we fix the parameter In view of Lemma 3.1 below, we therefore arrive at the estimate We have thus accomplished the proof of proposition.
Proof of lemma.We know that While by direct calculations it is seen that Furthermore, As an immediate consequence of the series of estimates established above, we obtain the important dissipative estimate for U at an exponential rate.Let us apply the estimates ( 15), ( 22) and (26) ] and [ 2t 3 , t], respectively.Then, it follows that where p(•)'s are continuous increasing functions determined in a suitable way.
Proposition 2. Let α 21 = 0.There exists a continuous increasing function p(•) such that, for any local solution U to (1) lying in (10) Proof.As before, a unified notation C will be used to denote various constants which are determined from the initial constants a, b, c, d, α ij (1 ≤ i, j ≤ 2) and γ ij (1 ≤ i, j ≤ 2) and the domain Ω alone in a specific way.So, C may change from occurrence to occurrence.When a constant C depends on a particular parameter, say ζ, we shall denote it by C ζ .Similarly, a unified notation p(•) will be used to denote various continuous increasing functions which may change from occurrence to occurrence.
Step 1. Integrate the equation for u of (1) in Ω.Then, Therefore, In addition, from As well, since it is seen that Step 2. Let q be an exponent varying in the range 2 < q < ∞.Multiply the second equation in (1) by qv q−1 and integrate the product in Ω.After some calculations, Here we notice, for example, that (indeed to see this, argue dividing the range of v into 0 ≤ v ≤ 2d/γ 22 and 2d/γ 22 < v < ∞).By this estimate it follows that Solving this differential inequality for Ω v q dx, we obtain that v(t) q Lq ≤ e −dqt v 0 q Lq + 2(2d/γ 22 ) q |Ω|, and hence Step 3. Multiply the second equation in (1) by d dt Q(v) = Q v v t and integrate the product in Ω.By obvious calculations, 1 2 In the meantime, multiply the second equation in (1) by Q(v) and integrate the product in Ω.After some calculations, Here we used (31).This differential inequality together with (36) then yields that Using (32), we can here prove the following lemma.
Lemma 3.2.There exists a constant C independent of the local solution such that Proof of lemma.We verify from (33) that where By (31) and (35) it is clear that Meanwhile, to estimate the integral of the function e −(t−s) N 2 (U (s)), we apply the second mean value theorem.Then there exists some τ ∈ [0, t] for which the formula Hence (38) is proved.
We have in this way concluded that As well, thanks to (34), integration of (37) on (0, t) yields that Step 4. In this step, we shall use the following abbreviated notation where p(•) is a continuous increasing function which varies in each occurrence.Introducing the quantity we intend to estimate N 1,log (u(t)) for the local solution.
Multiply the first equation in (1) by log(u + 1) and integrate the product in Ω.Then we can verify that by direct calculations utilizing the following formulae ∂u ∂t Here it is clear that it follows by ( 27) that Meanwhile, we have whatever the parameter ζ is, where C ζ denotes some constant determined from C ζ (and hence from ζ).Hence we obtain that Moreover, in view of ( 31) and (39), Let us add this differential equation to (37) and take ζ sufficiently small.Then it follows that where Step 5.In this step, we shall use the following abbreviated notation where p(•) is a continuous increasing function which varies in each occurrence.The goal is to estimate u(t) L2 for the local solution.
In the meantime, we have to prepare another differential inequality.Consider the duality product of the second equation of (1) and From (10) the equation for v has a meaning in the space By the Gagliardo-Nirenberg inequality, with any ζ 4 > 0. So, taking ζ 3 and ζ 4 sufficiently small and using (45), we obtain that 1 2 We now multiply a parameter η > 0 to (44) and add the multiplied inequality to (48).Then, If η is fixed in such a way that aη 2 ≥ p 2 (U 0 ) and if ζ 1 and ζ 2 are taken sufficiently small, then it is seen that Moreover, thanks to (39) and (42), L2 + 1) + 1}.We add this inequality to the following one 1 2 which is obtained from (36) (due to (42)) after some calculations and multiplication of a parameter ξ > 0. Then we arrive at the main differential inequality of this step Here we used also a fact that the estimate holds for all u ≥ 0, whatever the parameter ξ > 0 is, with some constant C ξ > 0.
Solving (49), we conclude that Moreover, by the corresponding versions of (34) and (41) obtained by integrating (30) and (41) on (s, t), respectively, we can verify that Here it is possible to fix the parameter ξ in such a way that Then it follows that ψ 2 (t) ≤ p 2 (U 0 ){e −t ψ 2 (0) + 1}.In particular, In view of (40), As well, thanks to (34), ( 41), ( 45) and (50), integration of (49) on (0, t) yields that Step 6.In this step, we shall use the following notation where p(•) is a similar continuous increasing function as before.The goal is to estimate u(t) L4 for the local solution.
Multiply the first equation in (1) by u 3 and integrate the product in Ω.Then, 1 4 Here, )dx with any ζ > 0 and any ζ ′ > 0. In addition, Here we used a fact that holds for all u ≥ 0, whatever the parameter ζ ′ > 0 is, where C ζ ′ > 0 is another constant depending on ζ ′ .We add this differential inequality to (49).Taking ζ ′ sufficiently small, we obtain that where ψ 3 (t) = ψ 2 (t) + 1 4 u(t) 4 L4 , for some positive exponent δ 1 > 0. Hence, In particular, Step 7. We shall use the following notation The goal is to estimate the norm u(t) H 1 for the local solution.
Multiply the first equation in (1) by d dt P (u, v) = P u u t + P v v t and integrate the product in Ω.By simple calculations, Here, thanks to (52), Furthermore, from ∇P (u, v) = P u ∇u + P v ∇v, it is immediate to see that In addition, (cf. ( 27)).Hence it is obtained that We add this differential inequality to (49).If the parameter ζ ′ is fixed sufficiently small, then where ψ 4 (t) = ψ 2 (t) + 1 2 ∇P (U (t)) 2 L2 , for some positive exponent δ 2 > 0. Hence, By (53), Thanks to (54), integration of (55) on (0, t) yields that Step 8. We shall use the following notation In view of ( 9), differentiate the first equation of (1) in t and consider the duality product with u t in H 1 (Ω) * × H 1 (Ω).After some calculations, 1 2 Here, with any ζ > 0. In addition, by Lemma 3.4 presented below, and, by a direct calculation, Similarly, by (59), Hence it is verified that 1 2 We add the previous differential inequality to (55) after multiplying both sides of (55) by a parameter ξ > 0. Then we arrive at the main differential inequality of this step . Solving this, we obtain that Moreover, by the corresponding versions of ( 51) and (58) obtained by integrating ( 49) and (55) on (s, t), respectively, we verify that If the parameter ξ is fixed in such a way that p 5 (U 0 ) − aξ ≤ −1, then In particular, Hence, by (59), we conclude that Thus, (61) together with (50) gives the desired estimate of proposition.
Proof of Lemma.We know by (54) that u H 2 ≤ C( ∆u L2 + u L2 ).By a direct calculation, we see that The desired estimate is then verified by the similar arguments as in the proof of Lemma 3.1.
As an immediate consequence of the series of estimates built up above, we obtain the dissipative estimate for U at an exponential rate.Let us apply (31) and (35) in 5.2.Compact absorbing set.On account of (65), for any 0 < R < ∞, there exists a suitable time t R > 0 such that This means that the estimate (82) in Section 6 is valid with C = p(2).We can then apply Theorem 6.4 to construct compact absorbing set.In fact, the set then X is a positively invariant set of S(t).In addition, X is a subset of K, is a bounded subset of H 2 (Ω), and is a compact set of X.For the detail, see the proof of Theorem 6.4.Thus, (S(t), X , X) defines a dynamical system.Every dynamical system of the form (S(t), B R , X) is reduced to it in the sense that S(t)B R ⊂ X 1 ⊂ X for t ≥ t R , t R being a suitable time depending on R.
in a Banach space X.Let Z be a second Banach space which is continuously embedded in X, and let K be an open ball of Z such that For each U ∈ K, A(U ) is a densely defined closed linear operator of X with a uniform domain D(A(U )) independent of U ∈ K.The operator F is a nonlinear operator from K into X.And, U 0 is an initial value at least in K.
We make the following structural assumptions, i.e., (68) ∼ (71).The spectral set σ(A(U )) is contained in a fixed open sectorial domain 2 , and the resolvent satisfies The domain D(A(U )) ≡ D is independent of U ∈ K, D being a Banach space with a graph norm • D = A(0) • X .The operator-valued function A(•) satisfies a Lipschitz condition of the form where Y is a third Banach space such that Z ⊂ Y ⊂ X with continuous embedding.The nonlinear operator F also satisfies a usual Lipschitz condition There are two exponents 0 ≤ α < β < 1 such that D(A(U ) α ) ⊂ Y and D(A(U ) β ) ⊂ Z for every U ∈ K with the estimates D i (i = 1, 2) being some constants independent of U ∈ K.
For the initial value U 0 ∈ K, we assume a compatibility condition of the form Furthermore, T U0 > 0 is determined by the norm A(U 0 ) γ U 0 X alone.
Let us finally verify Lipschitz continuity of solutions to (67) with respect to the initial values.For this purpose we introduce a set of initial values B = {U 0 ∈ Z; U 0 Z ≤ R 1 and A(U 0 ) γ U 0 X ≤ C 1 }, β < γ ≤ 1 with some constants 0 < R 1 < R and 0 < C 1 < ∞.Then, for each U 0 ∈ B, there exists a unique local solution.Moreover, by Corollary 1, we see that (67) possesses a local solution in the space given by (75) at least over an interval [0, T B ] for every initial value U 0 ∈ B.
We can then show the following theorem, see [2, Theorem 2].Theorem 6.2.Under (68) ∼ (71), let U and V be the local solutions to (67) with initial values U 0 and V 0 in the set B, respectively.Then, there exists some constant C B > 0 depending on the set B such that We assume that, for each R > 0, the family of linear operators A(U ), U ∈ K R , and the nonlinear operator F : K R → X satisfy all the structural assumptions (68) ∼ (71) announced in Subsection 6.1 with a third Banach space Y such that Z ⊂ Y ⊂ X and exponents 0 ≤ α < β < 1 which are all independent of R. If it is necessary for verifying (68) and (69), however, we may replace A(U ) (resp.F ) by A(U ) + k R (resp.F + k R ) in the equation of (76), where k R is some sufficiently large constant depending on R. Since We assume that there is a third exponent γ, where β < γ ≤ 1, such that holds.The space D γ is a Banach space with a graph norm • Dγ = A(0) γ • X .Let B be any bounded set of D γ , and take a semi-diameter R of K R sufficiently large in such a way that B ⊂ K R .By Corollary 1, for every U 0 ∈ B, there exists a unique local solution to (76) on a fixed interval [0, T B ], T B > 0 is determined by B. If we can show a priori estimates for all local solutions starting from B, then the global solutions are constructed.In fact, assume that there exist constants R B and C B such that the estimates

6 . 6 . 1 .
Quasilinear Abstract Parabolic Evolution Equations.Local problem.Consider the Cauchy problem for an abstract evolution equation  

6. 2 .
Global problem.On the basis of the results presented in the preceding subsection, we shall consider the global problem in a sense that the linear operator A(U ) is defined for every U ∈ Z and try to construct a global solution on the whole interval [0, ∞).Consider the Cauchy problem for an abstract parabolic evolution equation   dU dt + A(U )U = F (U ), 0 < t < ∞, U (0) = U 0 (76)in a Banach space X.For every U ∈ Z, A(U ) is a densely defined closed linear operator of X with a uniform domain D(A(U )) ≡ D, where Z ⊂ X is a second Banach space with continuous embedding.The domain D is a Banach space with a graph norm• D = A(0) • X .The operator F is a nonlinear operator from Z into X.And, U 0 is an initial value at least in Z.For 0 < R < ∞, let K R = {U ∈ Z; U Z < R}.

F
(U ) − A(U )U = {F (U ) + k R U } − {A(U ) + k R }U, U ∈ D,such replacement does not cause any essential change of equations in (76).