Solvability of the abstract evolution equations in L^s-spaces with critical temporal weights

1 Introduction

This paper deals with the abstract evolution equations in L s -spaces with critical temporal weights or equivalently critical L s -spaces. Given a Banach space X and a number 0 < T < ∞ , let L 0 ( 0 , T ; X ) be the collection of strongly measurable X -valued functions. Given two indices 0 < μ ≤ 1 and 1 < s < ∞ , the weighted abstract-valued L s -space and W k , s -space are defined as follows:

with the norm

and

with the norm

where f ( j ) denotes the j th derivative of f in the sense of distribution. Evidently, if μ = 1 in the aforementioned definitions, then we obtain the classical Lebesgue-Bochner space L s ( 0 , T ; X ) and Sobolev-Bochner space W k , s ( 0 , T ; X ) , respectively.

By means of Hölder’s inequality, Prüss-Simonett [1] showed that under the restriction μ > 1 / s , L μ s ( 0 , T ; X ) can be embedded into L 1 ( 0 , T ; X ) , hence the trace of W μ 1 , s ( 0 , T ; X ) makes sense. Moreover, the average operator

is bounded from L μ s ( 0 , T ; X ) to L μ s ( 0 , T ) . More precisely,

This is a direct corollary of Hardy’s inequality (cf. [1,2,3]).

Note that under the condition 1 / s < μ ≤ 1 , t ( 1 − μ ) s is a one-dimensional A s -weight of Muckenhoupt class, so L μ s ( 0 , T ; X ) can be used to investigate the maximal regularity of the abstract evolution equations. Suppose that A is a closed linear operator defined in X with the dense domain D ( A ) , and − A generates an analytic C 0 –semigroup e − t A . Under this assumption, A is called a sectorial operator (refer to [4, §2.5] or [5, p. 130]).

Consider the linear evolution equation:

We say (1.2) has the maximal L s -regularity on the interval [ 0 , T ] , or symbolically A ∈ M R s ( 0 , T ; X ) , if for every f ∈ L s ( 0 , T ; X ) , equation (1.2) has a unique strong solution u ∈ W 1 , s ( 0 , T ; X ) such that u ( 0 ) = 0 , A u ∈ L s ( 0 , T ; X ) , and (1.2) is satisfied for a.e. t ∈ ( 0 , T ] . Evidently, this strong solution u has the integral expression:

Moreover, by the closed graph theorem, there is a constant C ( s , T ) > 0 such that

From [6,7], we know that maximal L s -regularity is an essential property of A , it does not depend on the choice of s and T . In other words, if A ∈ M R s ′ ( 0 , T ′ ; X ) for some 1 < s ′ < ∞ and 0 < T ′ < ∞ , then A ∈ M R s ( 0 , T ; X ) for all 1 < s < ∞ and all 0 < T < ∞ . Moreover, the constant C ( s , T ) in (1.4) is uniform on [ s 0 , s 1 ] × ( 0 , T 0 ] for 1 < s 0 < s 1 < ∞ and 0 < T 0 < ∞ . For this reason, throughout this paper, we will omit the indices s , T , only use C to denote the regularity constant in (1.4), without specifying the bounds of s and T . For the sake of convenience, we also assume that semigroup e − t A is uniformly bounded, i.e., there is a constant M 0 ≥ 1 such that

Equation (1.2) is said to have the maximal L μ s -regularity on [ 0 , T ] , if for every f ∈ L μ s ( 0 , T ; X ) , (1.2) has a unique strong solution u ∈ W μ 1 , s ( 0 , T ; X ) satisfying u ( 0 ) = 0 and A u ∈ L μ s ( 0 , T ; X ) . With the aid of the boundedness of Φ on L μ s ( 0 , T ; X ) , [1] proved the equivalence of the maximal regularity of (1.2) on L s ( 0 , T ; X ) and on L μ s ( 0 , T ; X ) . This was also shown in [8] as a special case in the framework of singular operator theory.

There is a natural question arising up, that is whether or not the maximal regularity of (1.2) keeps in the critical case μ = 1 / s . To our best knowledge, there is not an affirmative answer to this question. In the method used in [1], average operator and Hardy’s inequality are employed, where the constant ( μ − 1 / s ) − 1 appears definitely. Since ( μ − 1 / s ) − 1 blows up as μ tends to 1 / s , this method fails here. Moreover, since t s − 1 is not a Muckenhoupt A s weight anymore, boundedness of the singular integral operator employed in [8] disappears in the space L 1 / s s ( 0 , T ; X ) . For these reasons, the question posed above seems much interesting, and it deserves full attention.

This paper makes a short investigation on the solvability of (1.2) in critical L s -spaces. Using decomposition of the integral solution (1.3), together with the weak boundedness of one-dimensional fractional integral operator on L 1 ( 0 , T ; X ) , we will show that if the function f belongs to L 1 ( 0 , T ; X ) ∩ L 1 / s s ( 0 , T ; X ) , then equation (1.2) has a strong solution in W loc 1 , 1 ( ( 0 , T ] ; X ) with A u , u ′ ∈ L 1 / s , w s ( 0 , T ; X ) . Furthermore, under the additional assumption Φ f ∈ L 1 / s s ( 0 , T ; X ) , then we also have A u , u ′ ∈ L 1 / s s ( 0 , T ; X ) , and u ∈ L ∗ s ( 0 , T ; X ) , an L s -space with the critical weight t − 1 . This gives a partial answer to the above question.

In the second part of the paper, solvability of (1.2) in critical L s -spaces, and estimates (1.5) are applied to the Cauchy problem of the semi-linear evolution equation:

where f : [ 0 , T ] × X → X is a Caratheodory map, locally Lipschitzian about u , and subject to a growth of the exponent − 1 w.r.t. t , i.e.,

and

for some positive function α ( t ) satisfying α ( t ) / t ∈ L 1 ( 0 , T ) and Φ ( α ( ⋅ ) / ⋅ ) ∈ L 1 / s s ( 0 , T ) . Using Banach’s contraction principle, with the aid of the function space L ∗ ρ , s ( 0 , T ; X ) (see Section 3), existence of the strong solution of equation (1.6) with u ( 0 ) = 0 is established. Since α ( t ) can be selected such that α ( t ) / t belongs to not L q ( 0 , T ) for any q > 1 , but L 1 ( 0 , T ) , results obtained here are useful supplements to the literature in dealing with nonlinear evolution equations despite that the initial value in our model is only 0.

For other investigations on the maximal L s -regularity of the evolution equations with different focuses, please refer to [9,10, 11,12] and [13, §3.4] etc.

2 Properties of critical L s -spaces and maximal L 1 / s s -regularity of evolution equations

Let 0 < T < ∞ and 1 < s < ∞ . Define the critical L s -space

Evidently, endowed with the norm

L 1 / s s becomes a Banach space. Furthermore, in the case s = 1 , L 1 / s s ( 0 , T ; X ) is exactly L 1 ( 0 , T ; X ) . As for s = ∞ , we define

with the norm

Unlike the subcritical case 1 / s < μ ≤ 1 , here L 1 / s s ( 0 , T ; X ) is not contained in L 1 ( 0 , T ; X ) for s > 1 anymore. For example, the function f ( t ) = ( t log t ) − 1 x ( x ∈ X ) belongs to L 1 / s s ( 0 , T ; X ) , but it does not belong to L 1 ( 0 , T ; X ) . On the other hand, direct calculation shows that

for all 1 < s < ∞ , and

for all f ∈ L 1 ( 0 , T ; X ) ∩ L 1 / ∞ ∞ ( 0 , T ; X ) . Furthermore, suppose that 1 ≤ s 1 < s < s 2 ≤ ∞ , and

for some 0 < θ < 1 , then we have

This shows the interpolation property of the critical L s -spaces. As a matter of fact, if we regard L 1 / s s ( 0 , T ; X ) as the space containing all the strongly measurable functions f satisfying

it is not hard to show that (cf. [14, §7.24] or[15, §5.2]), for every 1 < s < ∞ ,

in the sense of isomorphism. Here ( X , Y ) θ , s stands for the real interpolation space associated with the indices θ and s between the two Banach spaces X and Y .

Next we make a short investigation on the maximal regularity of evolution equation (1.2) in critical L s -spaces. Suppose that 1 < s < ∞ and f ∈ L 1 / s s ( 0 , T ; X ) , consider the decomposition of (1.3),

Let

Suppose that A ∈ ℳℛ ( 0 , T ) , then we have J 1 ∈ L 1 / s s ( 0 , T ; X ) . As for J 2 , by the estimates (1.5), we have

Since f ∈ L 1 ( 0 , T ; X ) , by the mapping property of the fractional integral operator (cf. [16, §6.2]), we can conclude that t 1 − 1 / s J 2 ∈ L w s ( 0 , T ; X ) , or J 2 ∈ L 1 / s , w s ( 0 , T ; X ) in symbol. Thus, J 1 + J 2 ∈ L 1 / s , w s ( 0 , T ; X ) , which implies that u ( t ) ∈ D ( A ) a.e. on ( 0 , T ] , and A u ∈ L 1 / s , w s ( 0 , T ; X ) . Consequently, u ′ ( t ) exists a.e. on ( 0 , T ] and u ′ ∈ L 1 / s , w s ( 0 , T ; X ) . Summing up, we have

For 1 < s < ∞ , define another type of weighted L s -space

with the norm

Here the temporal weight t − 1 is also critical, it does not belong to the Muckenhoupt class.

Let us derive some estimates for the solution of (1.6) for further arguments. First, using (1.3) and (1.5), we have

Thus, under the assumption Φ f ∈ L 1 / s s ( 0 , T ; X ) , we can conclude that t − 1 / s u ∈ L s ( 0 , T ; X ) , and

Moreover, if ρ ≥ s + 1 , then we have

It is easy to check that

defines a convex functional on L 0 ( 0 , T ; X ) . This functional also satisfies the following three properties

• λ ↦ φ ρ , s , T ( λ u ) is continuous on [ 0 , ∞ ) ,

• φ ρ , s , T ( λ u ) = φ ρ , s , T ( u ) provided ∣ λ ∣ = 1 , and

• φ ρ , s , T ( u ) = 0 if and only if u = 0 .

According to the definition of L ∗ ρ , s ( 0 , T ; X ) and inequality (2.6), we then obtain the second estimate of u ,

3 Solvability of semilinear evolution equations with temporal growth exponent − 1

Now, we are ready to deal with the semi-linear evolution equation (1.6). Here A is a sectorial operator having the maximal regularity on [ 0 , T ] with 0 < T < ∞ , F : [ 0 , T ] × X → X is a Caratheodory operator satisfying H ( F ) :

1. For all u ∈ X , t ↦ F ( t , u ) is strongly measurable on [ 0 , T ] ,

2. for a.e. t ∈ [ 0 , T ] , u ↦ F ( t , u ) is locally Lipschitz such that

   (3.1) ∥ F ( t , u ) − F ( t , v ) ∥ X ≤ C t ∥ u − v ∥ X ( α ( t ) + ∥ u ∥ X ρ − 1 + ∥ v ∥ X ρ − 1 )

   for some ρ ≥ s + 1 .

3. The one-dimensional Lebesgue measure of the set { t ∈ [ 0 , T ] : ∥ F ( t , 0 ) ∥ X > 0 } is larger than 0, and

   (3.2) ∥ F ( t , 0 ) ∥ X ≤ C α ( t ) t a.e. on [ 0 , T ] .

   Here α ( t ) ≥ 0 is a bounded function verifying

   (3.3) ∫ 0 T α ( t ) t d t < ∞ and Φ α ( ⋅ ) ⋅ ∈ L 1 / s s ( 0 , T ) .