# Quasilinear Evolution Equations inLμP-Spaces with Lower Regular Initial Data

Authors: Qinghua Zhang
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We study the Cauchy problem of the quasilinear evolution equations inLμp-spaces. Based on the theories of maximal Lp-regularity of sectorial operators, interpolation spaces, and time-weighted Lp-spaces, we establish the local posedness for a class of abstract quasilinear evolution equations with lower regular initial data. To illustrate our results, we also deal with the second-order parabolic equations and the Navier-Stokes equations in Lp,q-spaces with temporal weights.


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## Body

## 1. Introduction

In this paper, we will deal with the following quasilinear evolution equation:(1)u˙+At,uu=Ft,u,t>0,u0=u0in a Banach space couple (X,X1) with X1↪X densely. Take 1<p<∞ and μ∈(1/p,1], and denote by Xμ-1/p,p=(X,X1)μ-1/p,p the real interpolation space between X and X1 (see [1, Ch. 3] or [2, Ch. 1]). Given T0>0 or T0=+∞, let J0=[0,T0). And, for each 0<T<T0, let JT=[0,T]. Suppose that A:J0×Xμ-1/p,p→B(X1,X) is a continuous map, F:J0×X1-1/p,p→X is a Caratheodory map, and u0∈Xμ-1/p,p. We will deal with this problem in the framework of maximal regularity of the sectorial operators in Lp-spaces with temporal weights.Suppose thatA0 is a sectorial operator defined in X with the domain D(A0)=X1 (refer to [3–5] for the definition of sectorial operator), we say A0 has the maximal Lp-regularity on J0, or A0∈MRp(J0,X,X1) in symbol, and we mean that, for each f∈Lp(J0;X), the following Cauchy problem(2)u˙+A0u=ft,t>0,u0=0has a unique solution in W01,p(J0,X)∩Lp(J0,X1). And there is a constant C0>0, such that (3)u′LpJ0,X+uLpJ0,X1≤C0fLpJ0,Xfor u=(∂t+A^0)-1f with f∈Lp(J0,X). It is easy to see that if A0∈MRp(J0,X,X1), then A0∈MRp(JT,X,X1) for arbitrary T<T0. Especially, if A0∈MRp(R+,X,X1), then the constant C0 in (4) is independent of T.In general, a sectorial operatorA0 does not have the property of Lp-maximal regularity, even though its negative -A generates an exponentially decaying analytic semigroup. However, if X is a UMD space, and A0 admits bounded imaginary powers A0is (s≤ε for some ε>0), or A0∈BIP for short (see [3, Ch. 3], [5, 6]), then A0 has the Lp-maximal regularity. Further investigation, involving the vector-valued multiplier theorem and H∞-functional calculus of sectorial operators, showed that, in a UMD space, Lp-maximal regularity equals R-sectoriality for a sectorial operator (see [7], [8, Ch. 3]). Recall that, given Ω⊆Rn a bounded or exterior domain with compact C1 type boundary, Ω=Rn or a half space R+n, for each 1<q<∞, the Lebesgue space Lq(Ω) is of UMD type, on which the elliptic differential operators with some boundary conditions are R-sectorial (cf. [7, 9]). And the Stokes operator Ap is of BIP type with the power angle θAp<π/2 (see [10, 11]). Hence, we can make Lp-Lq estimates for the parabolic evolution equations (see [12, 13]) and the Navier-Stokes equations (see [14, 15]).As for the maximal regularity of a sectorial operatorA0 in Lμp-setting, or equivalently A0∈MRpμ(J0,X,X1), it means that, for each f∈Lμp(J0,X), Pr. (2) has a unique solution u∈W0μ1,p(J0,X)∩Lμp(J0,X1) with the estimate(4)u′LμpJ0,X+uLμpJ0,X1≤C0fLμpJ0,Xassociated with a μ-dependent constant C0. Prüss-Simonett in [16] showed that A0∈MRp(J0,X,X1) is equivalent to A0∈MRpμ(J0,X,X1) for all μ∈(1/p,1]. After that, Köhne-Prüss-Wilke in [17] studied Pr. (1) of autonomous type (i.e., A(t,u)=A(u) and F(t,u)=F(u)), including the local well posedness and long time behavior of the solutions, in Lμp-spaces.Motivated by the theory ofε-regular solutions (cf. [18, 19]), this paper will deal with Pr. (1) of nonautonomous type in weighted Lp-spaces. Unlike [17], here the regularity of the initial value is lower than that of the space where the nonlinearity F acts; more precisely, u0 lies in the space Xμ-1/p,p for some μ∈(1/p,1) and F maps J0×X1-1/p,p into X.This paper is organized as follows: as preliminaries, in Section2, we give some definitions and properties of the time-weighted Lebesgue and Sobolev spaces of abstract valued functions, and the maximal regular space E1,pμ(JT) and its closed subspace E01,pμ(JT). In Section 3, we establish the local and global existence of E1,pμ(JT) and Yμ,T solutions in subcritical and critical cases, respectively, using estimate (4), together with the embedding properties of E01,pμ(JT) and the smoothing action of the operator R. To illustrate the obtained results, in Section 4, we make Lμp-Lq estimates for the parabolic evolution equations and the Navier-Stokes equations, using the fractional order Sobolev space W2(μ-1/p),p(Ω) and Besov space Bq2(μ-1/p),p(Ω)n as the state spaces.Framework of our study can be incorporated into the theory of sectorial operators and analytic semigroups, together with time-weightedLp-spaces. Results obtained here have their meaning in the study of quasilinear evolution spaces driven by the sectorial operators. For the relative investigations on the quasilinear evolutions in Lp or Lμp spaces with different focuses, please refer to [5, 16, 17, 20], with the references therein.


## 2. Preliminaries

Let1<p<∞ and μ∈(1/p,1]; define(5)LμpJT,X=u∈LlocpJT,X:t1-μut∈LpJTwith the norm uLμp(JT,X)=t1-μuLpJT,X, and (6)Wμ1,pJT,X=u∈Wloc1,pJT,X:t1-μu′t∈LpJTwith the norm uWμ1,p(JT,X)=t1-μuW1,p(JT,X), and its closed subspace with zero trace (7)W0μ1,pJT,X=u∈Wμk,pJT,X:γu=0,where γu denotes the trace of u. We can also define the maximal regularity spaces (8)E1,pμJT=Wμ1,pJT,X∩LμpJT,X1,E01,pμJT=W0μ1,pJT,X∩LμpJT,X1.Here are some interpolating and embedding properties of the Lμp and Wμ1,p spaces, all of which can be found in [21, Ch. 1] or [22].(i)
Lμp(JT,X)↪Lq(JT,X) whenever 1≤q<p, and max⁡{1/p,1-1/p+1/q}<μ<1, and(9)uLqJT,X≤Tp-q/p-1-μq1-1-μpq/p-qp-q/puLμpJT,X.(ii)
For eachη>1-μ+1/p,(10)Wμη,pJT,X↪L∞JT,X,(iii)
For the indexesσ>0, q∈(1,∞), and θ∈(0,1),(11)Wμ1,pJT,LqΩ∩LμpJT,Wσ,qΩ↪Wμθ,pJT,Bqσ1-θ,pΩ.where Ω is a domain in Rn and Wσ,q(Ω) and Bqσ(1-θ),p(Ω)=(Lq(Ω), Wσ,q(Ω))1-θ,p denote the Sobolev and Besov spaces with real indexes, respectively.Suppose thatA0 is a sectorial operator with D(A0)=X1, and -A0 generates a uniformly bounded analytic semigroup e-tA0. From [4, Ch. 5], we know that, for each α∈(0,1) and all x∈Xα,p, the norm xXα,p is equivalent to(12)xX+∫0∞t1-αA0e-tA0xXpdtt1/p.Introduce the continuous function space with time weight(13)Cμ0,T;X1-1/p,p=u∈C0,T,X1-1/p,p:supt∈0,T⁡t1-μutX1-1/p,p<∞endowed with the norm(14)uCμ0,T;X1-1/p,p=supt∈0,T⁡utX1-1/p,pand its closed subspace(15)C0μ0,T;X1-1/p,p=u∈Cμ0,T;X1-1/p,p:limt→0+⁡t1-μutX1-1/p,p=0.Proposition 1.
E  0 1 , p μ ( J T ) ↪ C  0 μ ( 0 , T ; X 1 - 1 / p , p ) ∩ C 0 ( [ 0 , T ] ; X μ - 1 / p , p ) with the imbedding constant independent of T.Proof.
We will only proveE01,pμ(JT)↪C0μ(0,T;X1-1/p,p); for the other inclusion, please refer to [16]. Taking any u∈E01,pμ(JT), we have that Φμu∈E01,p(JT) and ΦμuE01,p(JT)≤CuE01,pμ(JT), where (Φμu)(t)=t1-μu(t) and the constant C is independent of T (see [21, Ch. 1]). Notice that E01,p(JT)↪C([0,T];X1-1/p,p) with the imbedding norm independent of T, and we then have t1-μu∈C([0,T];X1-1/p,p) with limt→0+⁡t1-μu(t)X1-1/p,p=0, and the desired result comes.Suppose that, for allu∈E01,pμ(JT),(16)max⁡uCJT,Xμ-1/p,uCμ0,T;X1-1/p≤C1uE1,pμJT,where C1>0 is independent of T.Proposition 2.
For eachu0∈Xμ-1/p,p, define (Ru0)(t)=e-tA0u0. Then Ru0∈C0ν(0,T;X1-1/p,p) for all ν∈1/p,μ together with the estimate(17)Ru0Cν0,T;X1-1/p,p≤CTTμ-νu0Xμ-1/p,p,where CT=max⁡{T1-μM,1} and M≥1 is the upper bounds of the semigroup e-tA0 on the infinite interval [0,∞).Proof.
Firstly, for eachν∈(1/p,μ], we can deduce that (18)t1-νe-tA0u0X1-1/p,p=t1-νe-tA0u0X+∫0∞A0e-s+tA0u0Xpds1/p≤tμ-νT1-μMu0X+∫t∞s1-μpA0e-sA0u0Xpds1/p≤CTtμ-νu0Xμ-1/p,p.Secondly, for each η∈(μ,1] and v∈Xη-1/p,p, we have that (19)limt→0+⁡t1-μe-tA0vX1-1/p,p=0.This relation, together with the dense imbedding X1-1/p,p↪Xμ-1/p,p and the upper bounds supt∈(0,T]⁡t1-μe-tA0u0X1-1/p,p≤u0Xμ-1/p,p, leads to the desired inclusion Ru0∈C0μ(0,T;X1-1/p,p).Analogously, we can derive the following.Proposition 3.
R u 0 ∈ E  0 1 , p ν ( J T ) and Ru0E1,pν(JT)≤CTTμ-νu0Xμ-1/p,p for all ν∈(1/p,μ] and u0∈Xμ-1/p,p, where CT=max⁡{21/p,M[(1-μ)p+1]-1/pT1-μ+1/p}.Remark 4.
From [5], we know that, for every sectorial operator A0 lying in MRp(R+,X,X1), the corresponding semigroup e-tA0 decays exponentially, and consequently the constant CT appearing in Propositions 1 and 2 is independent of T, whence the subscript T can be erased.The following lemma tells us that the property ofLp-maximal regularity can be preserved under small perturbations (refer to [5]).Lemma 5.
Given a sectorial operatorA0∈MRp(J0,X,X1) with the constant C0 as in (4), suppose that A1∈B(X1,X) is closed in X satisfying A1-A0B(X1,X)≤k/C0 for some κ∈[0,1). Then A1∈MRp(J0,X,X1) with the bounds C0/(1-κ).


## 3. Main Results and Proofs

Firstly, we give some hypotheses onA and F used in this paper.H ( A ): (i)A:J0×Xμ-1/p→B(X1,X) is continuous. And, for each R>0, there is a constant L(R)>0 such that (20)At,u-At,vBX1,X≤LRu-vXμ-1/p,pfor all t∈J0 and all u,v∈Xμ-1/p,p with uXμ-1/p,p,vXμ-1/p,p≤R.( i i ) A 0 = A ( 0 , u 0 ) ∈ M R p ( J 0 , X , X 1 ) for some u0∈Xμ-1/p,p.H ( F ): F:J0×X1-1/p,p→X is an operator of Caratheodory type; that is,( i ) for each u∈X1-1/p,p, F(·,u) is strongly measurable,( i i ) for a.a. t∈J0, F(t,·) is locally Lipschitz, and there is a number ρ>0 and a function ϕ∈L∞(J0,R+) for which (21)Ft,u-Ft,vX≤ϕtu-vX1-1/p,p1+uX1-1/p,pρ-1+vX1-1/p,pρ-1holds for a.a. t∈J0 and all u,v∈X1-1/p,p.,(iii)F(·,0)∈Lμp(0,T;X).It is easy to show that, under all the hypotheses listed above, the following functions associated withT: F(·,0)Lμp(0,T;X), maxt∈JT⁡A(t,u0)-A(0,u0)B(X1,X), Ru0Cμ(0,T;X1-1/p,p), maxt∈[0,T]⁡Ru0(t)-u0Xμ-1/p,p, and Ru0E1,pμ(JT) are all infinitely small as T→0. Without any confusion arising, in the coming discussions, the first small quantity is denoted by β(T), and other ones related to u0 are denoted by δ(T) uniformly.We shall now discuss the local well posedness of Pr. (1) in two cases.Case 1 (μ>1-p-1(ρ-1)-1).
For eachv∈E1,pμ(JT), consider the auxiliary problem(22)u˙+A0u=Ft,v-A0,v0-At,vv,t>0,u0=u1.
Introduce the mapG:E1,pμ(JT)→Lμp(JT,X): (23)Gv=F·,v-A0,u0-A·,vv.Then solution u of the auxiliary problem (22) can be represented by (24)u=Ru1+∂t+A^0-1Gv≔Fv,and consequently solution of the original problem (1) is exactly the fixed point of F.For anyu1∈Bu0μ(ε)={u∈Xμ-1/p,p:u-u0Xμ-1/p,p≤ε}, let (25)Bu0μr=u∈E1,pμJT:u0=u1,u-Ru1E1,pμJT≤r,where ε,r∈(0,1) are small positive numbers. Taking any v∈Bu0μ(r), using (16) and Proposition 2, we can derive that(26)vCJ,Xμ-1/p,p≤v-Ru1CJ,Xμ-1/p,p+Ru1-Ru0CJ,Xμ-1/p,p+Ru0CJ,Xμ-1/p,p≤C1v-Ru1E1,pμJT+Mu1-u0Xμ-1/p,p+CTu0Xμ-1/p,p≤C1r+Mε+CTu0Xμ-1/p,p≤C1+max⁡M,CT0+CT0u0Xμ-1/p,p≕R,and vCμ(0,T;X1-1/p,p)≤R in the same way. Now we can give the a priori estimates for F(·,v(·)) and (A(0,u0)-A(·,v))v in Lμp(0,T;X) as follows: (27)F·,v·Lμp0,T;X≤F·,0Lμp0,T;X+F·,v·-F·,0Lμp0,T;X≤βT+ϕ∞∫0Tt1-μpvtX1-1/p,pp·1+vtX1-1/p,pρ-1pdt1/p≤βT+ϕ∞T1/pvCμ0,T;X1-1/p,p+T1/p-1-μρ-11-1-μpρ-11/pvCμ0,T;X1-1/p,pρ≤βT+CϕT1/pR+T1/p-1-μρ-1Rρ,where Cϕ=ϕ∞max⁡{1,[1-(1-μ)p(ρ-1)]-1/p} and (28)A0,u0-A·,vvLμp0,T;X≤maxt∈0,T⁡A0,u0-At,u0BX1,XvLμp0,T;X1+A·,u0-A·,vvLμp0,T;X≤δT+LRmaxt∈0,T⁡vt-u0Xμ-1/p,pvLμp0,T;X1≤δT+LRC1r+Mε+δTvE1,pμJT≤δT+LRC1r+Mε+δTr+CT0ε+δT.Hence(29)Fv-R0u1E1,pμJT≤C0βT+CϕT1/pR+T1/p-1-μρ-1Rρ+δT+LRC1r+Mε+δTr+CT0ε+δT.In the next step, we will show the local Lipschitz property ofF. For this purpose, take any v,v¯∈Bu0μ(r) and derive that (30)F·,v·-F·,v¯·Lμp0,T;X≤ϕ∞∫0Tt1-μpvt-v¯tX1-1/p,pp1+vtX1-1/p,pρ-1+v¯tX1-1/p,pρ-1pdt1/p≤ϕ∞T1/p+T1/p-1-μρ-11-1-μpρ-11/pvCμ0,T;X1-1/p,pρ-1+v¯Cμ0,T;X1-1/p,pρ-1·v-v¯Cμ0,T;X1-1/p,p≤C1CϕT1/p+2T1/p-1-μρ-1Rρ-1v-v¯E1,pμJT,A0,u0-A·,vv-A0,u0-A·,v¯v¯Lμp0,T;X≤A0,u0-A·,vv-v¯Lμp0,T;X+A·,v-A·,v¯v¯Lμp0,T;X≤δTv-v¯Lμp0,T;X1+A·,u0-A·,vv-v¯Lμp0,T;X+LRv¯Lμp0,T;X1maxt∈0,T⁡vt-v¯tXμ-1/p,p≤δT+LRmaxt∈0,T⁡vt-u0Xμ-1/p,pv-v¯E1,pμJT+LRv¯E1,pμJTmaxt∈0,T⁡vt-v¯tXμ-1/p,p≤δT+2C1LRr+M+C1CT0LRε+1+C1LRδTv-v¯E1,pμJT.Consequently(31)Fv-Fv¯E1,pμJT≤C0C1CϕT1/p+T1/p-1-μρ-1Rρ-1+δT+2C1LRr+M+C1CT0LRε+1+C1LRδTv-v¯E1,pμJT.Take(32)r≤min⁡22+C01+LRC1+M+1-1,4C01+3C1+M+CT0C1+1LR-1and ε≤r, select T1∈(0,T0) such that βT1≤r/4C0 and δT1≤r, and let(33)T=min⁡T1,r8C0CϕRρ1/1/p-1-μρ-1,and then, from (29) and (31), we can deduce that, for all u1∈Bu0μ(ε) and v,v¯∈Bu0μ(r), Fv-R0u1E1,pμ(JT)≤r and Fv-Fv¯E1,pμ(JT)≤2-1v-v¯E1,pμ(JT). Thus by Banach’s contraction theorem, the nonlinear map F has a unique fixed point, which is the unique local solution of Pr. (1) simultaneously.Theorem 6.
Suppose all the hypotheses uponF and A together with μ>1-p-1(ρ-1)-1 are satisfied, and then there exist ε>0 and T>0 such that, for each u1∈Bu0μ(ε), Pr. (1) has a unique solution in the space E1,pμ(JT).Now, we turn to deal with the continuation of the local solutions. Suppose thatH(A) (ii) is strengthened as follows:( i i ′ ) For all (t,u)∈J0×Xμ-1/p,p, A(t,u)∈MRp(R+).Under this situation, by invoking Lemma5, together with the continuity of A, we can deduce that, for each precompact subset K⊂J0×Xμ-1/p,p, there is a common constant C0, for which Ine. (4) is satisfied for all A(t,u) with (t,u)∈J0×K. We can also assume that all the semigroups e-sA(t,u) share the same bounds M.Taking anyu0∈Xμ-1/p,p, and using Theorem 6, we can find a finite interval JT⊆J0, on which Pr. (1) has a unique solution u~∈E1,pμ(JT) with the initial value u~(0)=u0. Denote by A¯(t,u)=A(t+T,u) and F¯(t,u)=F(t+T,u). Evidently, A¯,F¯ verify hypotheses H(A) and H(F), respectively. Notice that u~∈E1,p([t¯,T]) for arbitrary 0<t¯<T. T, and then, following the same process as in [17], we can find a time τ∈(T,T0] for which there is a unique solution u¯ surviving on the maximal existing interval [0,τ-T) solving the following problem: (34)u¯˙+A¯t,u¯u¯=F¯t,u¯,t>0,u¯0=u~T.Define(35)ut=u~t,t∈0,T,u¯t-T,t∈T,τ.Then u∈E1,pμ(T~) for any 0<T~<τ and solves (1) on the interval [0,τ). We can also conclude that u could not be extended beyond τ anymore. Thus the function u constructed above is exactly the solution of (1) arising out of u0∈Xμ-1/p,p with the maximal existing interval [0,τ). If τ=T0, then u is called globally existing. If τ<T0, then we obtain (36)∫0τt1-μpu˙tXp+utX1pdt=∞,and limt→τ-u(t) does not exist in Xμ-1/p,p. In this case, we say that u blows up at time τ.Theorem 7.
Suppose thatX1↪↪X, and there exists ν∈(1-p-1(ρ-1)-1,μ) such that H(A) (i) is also satisfied with ν. Suppose also, on its maximal interval of existence [0,τ), solution u of Pr. (1) verifies (37)supt∈0,τ⁡utXμ-1/p,p<∞.Then τ=T0, and u survives globally.Proof.
We will prove it by contradiction. Suppose thatτ<T0, and then the set K={(t,u(t)):t∈[0,τ)} is bounded in J0×Xμ-1/p,p and therefore precompact in J0×Xν-1/p,p since X1↪↪X. Thus uτ=limt→τ-⁡u(t) exists in Xν-1/p,p. Define u(τ)=uτ, and then u can be extended continuously onto the whole interval [0,τ]. Assume that supt∈[0,τ]⁡u(t)Xν-1/p,p=Rν, and then (38)At¯+δ,ut¯+δ-At¯+δ+t,ut¯+δBX1,X≤2LRνut¯+δ-ut¯Xν-1/p,p+At¯,ut¯-At¯+t,ut¯BX1,X.Thus, by the continuity of u and A, we can deduce that (39)limT→0⁡supt∈0,T⁡At¯+t,ut¯-At¯,ut¯BX1,X=0uniformly for t¯∈[0,τ].
In addition, for arbitraryε>0, there is a partition of [0,τ]: 0=t0<t1<⋯<tm=τ such that (40)∫0Tt1-νpFt+t¯,0-Ft+s¯,0Xpdt<εprovided t¯,s¯∈[tk-1,tk] for all k∈{1,2,…,m}. Notice that (41)limT→0⁡∫0Tt1-νpFt+tk,0Xpdt=0for all k∈{1,2,…,m}; we obtain (42)limT→0⁡∫0Tt1-νpFt+t¯,0Xpdt=0uniformly for t¯∈[0,τ].
Now we can take an arbitrary sequence{tn} converging to τ from the left side, and consider the following problem:(43)u˙n+A¯nt,unun=F¯nt,un,t>0,un0=utn,where A¯n(t,u)=A(t+tn,u) and F¯n(t,u)=F(t+tn,u). Assume that (16) holds also with ν, and then, by repeating the reasoning process as in Theorem 6  n times with the same C0, C1, β(T), and δ(T), we obtain a small number T>0, such that, on the common interval [0,T], Pr. (43) has a unique solution un in E1,pν(JT). By the uniqueness, we can find that u(t)=un(t-tn) for t∈[tn,tn+T], which means that u can be extended beyond the singular point τ. This contradicts to the maximality of τ since tn+T>τ for n large enough. Thus the proof has been completed.Case 2 (μ=1-p-1(ρ-1)-1((p-1)(ρ-1)>1)).
In this case, we only deal with the Cauchy problem of the semilinear parabolic equation; namely,(44)u˙+A0u=Ft,u,t>0,u0=u0.Here, A0 is a sectorial operator lying in MRp(R+).Fixν∈(1/p,μ) and 0<T≤1, and suppose that inequalities (4) and (16) also hold for A with μ replaced by ν. Take (45)Yμ,T=C0,T;Xμ-1/p,p∩Cμ0,T;X1-1/p,pas our work space, which is a Banach space endowed with the norm (46)vYμ,T=max⁡vC0,T;Xμ-1/p,p,vCμ0,T;X1-1/p,p,and let Y0μ,T=C0([0,T];Xμ-1/p,p)∩C0μ(0,T;X1-1/p,p). Consider the ball in Yμ,T: (47)Bu1μr=v∈Yμ,T:v-Ru1∈Y0μ,T,v-Ru1Yμ,T≤r.For all u1∈Bu0μ(ε), v,v¯∈Bu1μ(r), and t∈(0,T], by H(F), we can deduce that (48)F·,v·Lνp0,t;X≤tμ-νF·,0Lμp0,t;X+ϕ∞vLνp0,t;X1-1/p,p+∫0ts1-νpvsX1-1/p,pρpds1/p≤tμ-νβt+ϕ∞tμ-ν+1/pμ-νp+11/pvCμ0,t;X1-1/p,p+∫0tt1-νp-1-μpρdt1/pvCμ0,t;X1-1/p,pρ≤tμ-νβt+Cϕ,νt1/pv-Ru1Cμ0,t;X1-1/p,p+Ru1Cμ0,t;X1-1/p,p+v-Ru1Cμ0,t;X1-1/p,p+Ru1Cμ0,t;X1-1/p,pρ,where Cϕ,ν=ϕ∞[p(μ-ν)]-1/p. And (49)F·,v·-F·,v¯·Lνp0,t;X≤ϕ∞t1/p+μ-νμ-νp+11/p+tμ-νpμ-ν1/pvCμ0,t;X1-1/p,pρ-1+v¯Cμ0,t;X1-1/p,pρ-1·v-v¯Cμ0,t;X1-1/p,p≤Cϕ,νtμ-νt1/p+2r+Cε+δtρ-1v-v¯Yμ,t.On the other hand, for each w∈E01,pν(JT), we have (50)wtXμ-1/p,pp=wtXp+∫0∞s1-μpA0e-sA0wtXpds≤wtXp+t1-μp∫0tA0e-sA0wtXpds+tν-μp∫t∞s1-νpA0e-sA0wtXpds≤tν-μpt1-νpwtX1-1/p,pp+wtXν-1/p,pp≤2C1ptν-μpwE1,pνJTp.Therefore, for the corresponding function u=Ru1+(∂t+A0)-1∘Gv, we obtain (51)max⁡ut-Ru1tXμ-1/p,p,t1-μut-Ru1tX1-1/p,p≤2C0C1βt+Cϕ,νt1/pv-Ru1Cμ0,t;X1-1/p,p+Ru1Cμ0,t;X1-1/p,p+v-Ru1Cμ0,t;X1-1/p,p+Ru1Cμ0,t;X1-1/p,pρfor all t∈(0,T], which means that u-Ru1∈Y0μ,T, and(52)u-Ru1Yμ,T≤2C0C1βT+Cϕ,νT1/pr+ε+δT+r+ε+δTρ.Furthermore, for another function u¯=(∂t+A0)-1v¯, we have(53)u-u¯Yμ,T≤2C0C1Cϕ,νT1/p+2r+ε+δTρ-1v-v¯Yμ,T.Take(54)r≤8C0C1Cϕ,νC+2ρ-1/ρ-1and ε≤r, select T1∈(0,T0) such that βT1≤r/4C0C1 and δT1≤r, and let(55)T=min⁡T1,8C0C1Cϕ,νC+2-p,and then, from the a priori estimates (52) and (53), we can derive that, for all u1∈Bu0μ(ε) and v,v¯∈Bu0μ(r), (56)∂t+A^0-1∘GvYμ,T≤r,∂t+A^0-1∘Gv-∂t+A^0-1∘Gv¯Yμ,T≤2-1v-v¯Yμ,T.Thus using Banach’s contraction theorem, we can conclude that the multiple map Ru1+(∂t+A^0)-1∘G has a unique fixed point, which is the unique local solution of Pr. (44) exactly. That is, consider the following theorem.Theorem 8.
Assume that hypothesesH(F) are satisfied; then there exist ε>0 and T>0, such that, for each u1∈Bu0μ(ε), Pr. (44) has a unique solution in the space Yμ,T.Remark 9.
Like Case1, each Yμ,T solution of Pr. (44) can be extended onto a maximal interval [0,τ) such that limt→τ-⁡u(t) does not exist in Xμ-1/p,p provided τ<∞. Moreover if X1↪↪X, and there exists μ1>1-1-p-1(ρ-1)-1 such that (57)supt∈0,τ⁡utXμ1-1/p,p<∞,then this solution exists globally.


## 4. Examples

Example 10.
Consider the second-order parabolic equation(58)∂tu-∑i,j=1naijt,x,uDiju=ft,u,∇u,t>0,x∈Ω,u∂Ω=0,t>0,u0=u0,x∈Ω.Suppose that Ω⊆Rn (n≥3) is bounded domain with C2 boundary, 1<p<∞, J0=[0,T0] with T0<∞, and aij∈C(J0×Ω¯×R),i,j=1,2,…,n, and (aij(t,x,u))n×n is a symmetric and positive definite matrix for every (t,x,u)∈J0×Ω¯×R. Suppose also f:J0×R×Rn→R is a function of Caratheodory type; that is, t↦f(t,u,ξ) is measurable for each (u,ξ)∈R×RN, and (u,ξ)↦f(t,u,ξ) is locally Lipschitz for a.e. t∈J0. Moreover, there are numbers ρ0,ρ≥1 and a function ϕ∈L∞(J0), such that (59)ft,u1,ξ1-ft,u2,ξ2≤ϕtu1-u21+u1ρ0-1+u2ρ0-1+ξ1-ξ21+ξ1ρ-1+ξ2ρ-1for a.e. t∈J0, all ui∈R, and all ξi∈Rn,i=1,2.LetX=Lp(Ω) and X1={u∈W2,p(Ω):u∂Ω=0}, where u∂Ω is the trace of u on ∂Ω. Then the interpolate space Xμ-1/p,p can be described as (60)Xμ-1/p,p=u∈W2μ-2/p,pΩ:u∂Ω=0,μp>3/2,W2μ-2/p,pΩ,1<μp<3/2.By Nirenberg-Gagliardo’s inequality, we have that Xμ-1/p,p↪C(Ω¯) provided(61)p>n+22,μ>n+22p,and W1-2/p,p(Ω)↪Lp(ρ-1)(Ω) whenever(62)1ρ-1+p-2n≥1.Therefore, for each R>0, there is correspondingly a constant CR>0 such that, for any ui∈X1-1/p,p with uiC(Ω¯)≤R,i=1,2, we have (63)Ft,u1-Ft,u2X≤ϕtu1-u2LpΩ+∇u1-∇u2LpΩ·1+Rρ0-1+∇u1Lpρ-1Ωρ-1+∇u2Lpρ-1Ωρ-1≤CRϕtu1-u2X1-1/p,p1+u1X1-1/p,pρ-1+u2X1-1/p,pρ-1,where F(t,ui)=f(t,ui,∇ui),i=1,2.Givenv∈Xμ-1/p,p and u∈X1, define (64)At,vux=-∑i,j=1naijt,x,vxDijux.It is easy to show that A∈C(J0×Xμ-1/p,p,B(X1,X0)), and, from [7, 12], we know that A(t,v)∈MRp(J0,X0) for all (t,v)∈J0×Xμ-1/p,p.Now the numbersp,μ can be chosen to fit in our setting according to the variation of ρ. In fact, if 1≤ρ≤2, then we take (65)n+22<p≤min⁡n+22+1ρ-1,θn,μ>n+22p;if 2<ρ≤2n/(n+2)+1, then we take(66)n+22<p≤min⁡n+22+n-1ρ-1,θn,μ>n+22p;if ρ>2n/(n+2)+1, then we take(67)2+n1-1ρ-1≤min⁡p≤n+2+n-1ρ-1,θn,μ>n+22p,where 0<θ<1 is arbitrary. In this setting, all Conditions (61), (62), μ>1-1/[p(ρ-1)], and n/2p<1/2 are verified. Thus, using Theorems 6 and 7, we have the following.Theorem 11.
For the numbersp,μ chosen above, if the initial point u0∈Xμ-1/p,p (including the case X1/2,p=W01,p(Ω)), there is a radius ε>0 and a time T>0 depending only on the Xμ-1/p,p-norm of u0, such that, for each u1∈Bu0μ(ε), there is a unique solution u=u(·,u1) to Pr. (58) on the interval [0,T] with u(0)=u1 and satisfying (68)u∈Wμ1,p0,T;LpΩ∩Lμp0,T;W2,pΩ∩C0,T;C0Ω∩Cμ0,T;W2-2/p,pΩ.And, for any two points ui,i=1,2, lying in the same ball Bu0μ(ε), and w(t)=u(t,u1)-u(t,u2), we have the following estimates: (69)∫0Tt1-μpwtW2,pΩpdt+∫0Tt1-μp∂twtLpΩpdt1/p≤Cu1-u2Xμ-1/p,p,max⁡maxt∈0,T⁡wtCΩ,supt∈0,T⁡t1-μ∇wtLpρ-1Ω≤Cu1-u2Xμ-1/p,pfor some C>0.
Any such solutionu can be extended onto a maximal interval [0,T) uniquely. Furthermore, if (70)∫0τt1-μputW2,pΩpdt+∫0τt1-μp∂tutLpΩpdt1/p<∞or weakly supt∈[0,τ)⁡u(t)Xμ-1/p,p<∞, then τ=T0, and the solution u survives globally.Remark 12.
Since the regularity of the initial value is allowed to be lower than that of the space where the nonlinear mapF(t,·) acts, here we need not require W1-2/p,p(Ω)↪C(Ω¯), and, consequently, condition p>n+2 which was used in [17] is replaced by the weaker one: p>(n+2)/2 in this paper.Example 13.
Let us consider the homogeneous Navier-Stokes equation(71)ut-νΔu+u·∇u+∇p=ft,t>0,x∈Ω,div⁡u=0,t>0,x∈Ω,u∂Ω=0,t>0,u0=u0,x∈Ω.HereΩ⊆R3 is a bounded domain with C2+η (η>0) boundary. The vector-valued function u denotes the velocity of the flow, and the scalar function p denotes the pressure, ν is the viscous coefficient, and f is the body force. Denote (72)Dσ=Ψ∈C0∞Ω3:div⁡Ψ=0,and let Hq be the complement of Dσ in the space Lq(Ω)3. Recall that, under the assumptions upon Ω, we have that (73)Hq=u∈LqΩn:div⁡u=0,γnu=0,where div⁡u is the divergence of u in the sense of distribution, and γnu denotes the trace of u such that γnu=u·n for all u∈C1(Ω¯)3, where n denotes the unit outer normal vector at x∈∂Ω. Moreover we have the Helmholtz decomposition (see [23, Ch. III]) (74)LqΩ3=Hq⊕GqwithGq=∇p:p∈W1,qΩ,from which we obtain a continuous projection Pq:Lq(Ω)3→Hq which is called the Helmholtz projection. Define (75)DAq=u∈W2,qΩ3:div⁡u=0,inΩ,u∂Ω=0,Aqu=-PqΔu.Recall that, for each 1<q<∞, Aq is a sectorial operator of BIP type in Hq, and consequently it has the Lp-maximal regularity on the infinite interval R+ (cf. [10]).LetB(u,u)≔Pq(u·∇)u; then (71) can be abstracted by(76)ut+νAqu+Bu,u=Pqft,t>0,u0=u0.For each θ∈(0,1), denote Dqθ,p≔(Hq,D(Aq))θ,p; then we obtain Dqμ-1/p,p↪Bq2μ-2/p,p(Ω)3. Notice that (cf. [24, Ch. 7]), for every s>0,(77)Bqs,pΩ↪CΩ¯,providedsq>3,LαΩ,providedsq=3,α∈1,∞,Lη,pΩ↪LηΩ,providedsq<3,η=3q3-sq>p,where Lβ,p(Ω) denotes the Lorenz space. And (78)Bqs,pΩ↪W1,qΩ,provideds>1,Bq1,2Ω↪W1,qΩ,provided2≤q<∞.From the above embedding relations, we can derive the a priori estimates for the nonlinear term (u·∇)u as follows:(1)
Ifp≥2, q>3[2(1-1/p)]-1, then (79)u·∇uLqΩ≤uL∞Ω∇uLqΩ≤CuBq2-2/p,pΩ2.(2)
Ifp>2, (1-4/(3p))-1<q≤3[2(1-1/p)]-1, then (80)u·∇uLqΩ≤uLqrΩ∇uLqr′Ω≤CuBq2-2/p,pΩ∇uBq1-2/p,pΩ.Here 1-2q(1-1/p)/3<r<(1-2/p)/3 and 1/r+1/r′=1. Therefore the nonlinear map B(·,·) verifies (81)Bu,uHq≤CuDq1-1/p,p2,Bu1,u1-Bu2,u2Hq≤Cu1-u2Dq1-1/p,pu1Dq1-1/p,p+u2Dq1-1/p,pprovided(82)p>2,43p+1q<1.Pick μ∈(1-1/p,1]; then all the assumptions of H(F) for the operator F(t,u)=B(u,u)+Pqf(t) are satisfied with ρ=2 provided f∈Lμp(R+,Lq(Ω)3). Thus, using Theorems 6 and 7 for the semilinear equations, we can derive the following.Theorem 14.
For eachu0∈Dqμ-1/p,p and f0∈Lμp(R+,Lq(Ω)3), there exist ε,T>0 depending on the norm of u0Dqμ-1/p,p and f0Lμp(R+,Lq(Ω)3), such that, for every u1∈Bu0μ(ε) and f∈Lμp(R+,Lq(Ω)3) with f-f0Lμp(R+,Lq(Ω)3)≤ε, Pr. (76) has a unique solution u=u(·,u1,f) on the interval [0,T] with the initial condition u(0)=u1 verified. This solution satisfies (83)u∈Wμ1,p0,T;LqΩ3∩Lμp0,T;W2,qΩ3∩C0,T;Bq2μ-2/p,pΩ3∩Cμ0,T;Bq2-2/p,pΩ3.And for all ui∈Bu0μ(ε) and fi∈Lμp(R+,Lq(Ω)3) with fi-f0Lμp(R+,Lq(Ω)3)≤ε,i=1,2, by denoting w(t)=u(t,u1,f1)-u(t,u2,f2), we have that (84)∫0Tt1-μpwtW2,qΩpdt+∫0Tt1-μp∂twtLqΩpdt1/p≤Cu1-u2Dqμ-1/p,p+f1-f2LμpR+,LqΩ3,max⁡maxt∈0,T⁡wtBq2μ-2/p,pΩ,supt∈0,T⁡t1-μ∇wtLqΩ,supt∈0,T⁡t1-μwtCΩ≤Cu1-u2Dqμ-1/p,p+f1-f2LμpR+,LqΩ3for some C>0.
If we denote by[0,τ) the maximal interval of existence of the solution u=u(·,u0,f) to Pr. (76) extended uniquely by the local solution, then τ<∞ with (85)∫0τt1-μputW2,qΩpdt+∫0τt1-μp∂tutLqΩpdt1/p=∞and supt∈[0,τ)u(t)Dqμ-1/p,p=∞, or τ=∞ and the solution u exists globally.
Furthermore, for eachT∈(0,τ), we have the following estimate:(86)∫0Tt1-μputW2,qΩpdt+∫0Tt1-μp∂tutLqΩpdt1/p≤CTu0Dqμ-1p,p+fLμpR+,LqΩ3for some CT>0.Remark 15.
Iff∈Lνp(R+,Lq(Ω)3) for some 1-1/p<ν<μ, then we can also derive that u∈C0μ(0,T;Dq1-1/p,p); that is,(87)limt→0+⁡t1-μutDq1-1/p,p=0.Let us consider a special case. Letq=2, 8/3<p<4, and 1-1/p<μ<1/2+1/p, and then we have (88)β1=μ-1p∈14,12,γ1=1-1p∈58,34,ε=1-μ∈18,38.So if we take β1<β and γ1>γ, then we have D(A2β)↪D2βγ1,p and D2γ1,p↪D(A2γ). Thus, for u0∈D(A2β), we obtain limt→0+tεu(t)D(A2γ)=0. Compared to limt→0+⁡tβu(t)D(A2γ)=0 which was derived in [25], despite the fact that the upper bounds of the regularity of D(A2γ) is lower than 3/4, but, as a compensation, the lower bounds of exponent ε are less than 1/4, which means that, here, solution u(t) exhibits a better time asymptotic property as t→0. Notice that ε=1-μ=(1-1/p)-(μ-1/p), and limit (87) turns to be limt→0+⁡tεu(t)Dqβ+ε,p=0. This is an Lq-version of the limit limt→0+⁡tεu(t)D(A21/4+ε)=0 (u0∈D(A21/4), ε∈[1/4,3/8]) which was derived in [19] by means of ε-regular solution. It is worth pointing out that, by using the maximal Lμp-regularity of the Stokes operator Aθ, the continuity assumptions of the forcing term f has been removed here. Moreover, the asymptotic regular assumption upon f as t→0: P2f(t)D(A22ε-3/4)=o(t-2ε) which was used in [19] or P2f(t)H2=o(t-1+β) which was used in [25] are replaced by a much more relaxed one: f∈Lνp(R+,Lq(Ω)3). In this sense, results obtained here can be regarded as useful supplements of those in [19, 25] for the case q≠2.



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*Source: 2569080-2018-05-02.xml*