Fractional nonlinear heat equations and characterizations of some function spaces in terms of fractional Gauss–Weierstrass semi–groups

We present a new proof of the caloric smoothing related to the fractional Gauss–Weierstrass semi–group in Triebel-Lizorkin spaces. This property will be used to prove existence and uniqueness of mild and strong solutions of the Cauchy problem for a fractional nonlinear heat equation. Math Subject Classiﬁcations. 35K30, 35K05, 46E35


Introduction
The aim of the paper is twofold.First we justify the smoothing property of the fractional Gauss-Weierstrass semi-group W α t , in Besov and Triebel-Lizorkin spaces (see Definition 2.1) A s p,q (R n ), A ∈ {B, F }, s ∈ R and 1 ≤ p, q ≤ ∞. (1.3) Here ∧ and ∨ stand for the Fourier transform and its inverse, respectively.It will be a straightforward consequence of the characterization of some of these spaces in terms of the semi-group W α t .Based on these observations we deal secondly with the Cauchy problem where 0 < T ≤ ∞ and 2 ≤ n ∈ N in the context of the semi-group W α t in (1.2) and the fractional Laplacian (−∆) α w = |ξ| 2α w ∨ . (1.6) Here as usual ∂ t = ∂/∂t and ∂ j = ∂/∂x j .In particular, (−∆) α = (−∆ x ) α refers to the space variables.If α = 1 then (1.4) refers to Burgers equation.The peculiar nonlinearity Du 2 = n j=1 ∂ j u 2 is considered as the scalar counterpart of the related (vector-valued) non-linearity in the Navier-Stokes equations.In general the fractional Laplacian (−∆) α is modelling dissipation (hyperdissipation if α > 1, hyperviscousity if α = 2).In this respect the case α = n+2 4 attracted special attention.We refer to [7] and [13].The fractional Burgers equation (1.4) with 1/2 ≤ α < 1 has been considered in [6].For investigations of solutions of Cauchy problems for fractional dissipative heat equations with several types of nonlinearities we refer to [12].It turns out that both in generalized Navier-Stokes equations and in other generalized equations of physical and biological relevance (such as quasi-geostrophic equations, Keller-Segel equations, chemotaxis equations) the suggestion is to replace the Laplace operator by a (fractional) power (−∆) α in order to achieve at adequate mathematical models.The motivation to study the smoothing property (1.1) for fractional Gauss-Weierstrass semi-groups comes from our interest in so-called mild solutions of (1.4), (1.5) being fixed points of the operator T u 0 , in suitable weighted Lebesgue spaces with respect to the Bochner integral L v (0, T ), b, X), where X = A s p,q (R n ), A ∈ {B, F }, s ∈ R, 1 ≤ p ≤ ∞, 1 ≤ q ≤ ∞, stands for a Besov or Triebel-Lizorkin space.This means that for some b ∈ R and v our solution satisfies Note that (after extension from R n × (0, T ) to R n+1 by zero) (see [18, formulae (4.17), (4.21)]) for details and an appropriate interpretation).
Here S ′ (R n+1 ) stands for the space of tempered distributions.As far as the smoothing property (1.1) is concerned we refer to [18,Theorem 4.1] (α = 1, classical case), [2, Proposition 3.4] (α ∈ N) as well as [8,Corollary 5.4] and [9,Example 4.7] (α > 0).We present a different proof in Theorem 2.7 based on characterizations of A s p,q (R n ) (s > 0) in terms of the fractional Gauss-Weierstrass semi-group W α t provided in Theorem 2.5.As far as the Cauchy problem (1.4), (1.5) is concerned we follow the method developed in [18, Subsection 4.5] and [1] (α = 1) as well as [2] and [3] (α ∈ N).The main results are contained in Theorem 3.4 (existence of mild and strong solutions) and Theorem 3.5 (locally well-posedness of the Cauchy problem).Our approach allows us to deal with the above Cauchy problem for initial data u 0 belonging to spaces A s 0 p,q (R n ) in the so-called supercritical case for spatial solution spaces The paper is organized as follows.Section 2 is concerned with the characterization of spaces A s p,q (R n ) in terms of fractional Gauss-Weierstrass semi-groups and the proof of the smoothing property (1.1).The existence, uniqueness and stability of mild and strong solutions of the Cauchy problem (1.4), (1.5) are treated in Section 3. In the final Section 4 we illustrate different cases of solution spaces depending on the choice of α > 0, dimension n and integrability p.In particular we consider how close the spaces of admitted initial data approach to the so-called critical line s 0 = n p − 2α + 1 (for more details see explanations below).Moreover, we discuss the special case when initial data u 0 belong to L p (R n ), 1 < p < ∞.Finally, we investigate how our results fit in the current literature and compare them with related results.

Function spaces 2.1 Definitions and basic ingredients
We use standard notation.Let N be the collection of all natural numbers and N 0 = N ∪ {0}.Let R n be Euclidean n-space, where n ∈ N. Put R = R 1 .Let S(R n ) be the Schwartz space of all complex-valued rapidly decreasing infinitely differentiable functions on R n and let S ′ (R n ) be the space of all tempered distributions on R n .Furthermore, L p (R n ) with 0 < p ≤ ∞, is the standard complex quasi-Banach space with respect to the Lebesgue measure, quasi-normed by with the usual modification if p = ∞.Similarly L p (M ) where M is a Lebesguemeasurable subset of R n .As usual, Z is the collection of all integers and Z n where n ∈ N denotes the lattice of all points m = (m 1 , . . ., m n ) ∈ R n with m k ∈ Z.Let Q j,m = 2 −j m + 2 −j (0, 1) n with j ∈ Z and m ∈ Z n be the usual dyadic cubes in R n , n ∈ N, with sides of length 2 −j parallel to the axes of coordinates and 2 −j m as the lower left corner. If denote the Fourier transform of ϕ.As usual, F −1 ϕ and ϕ ∨ stand for the inverse Fourier transform, given by the right-hand side of (2.2) with i in place of −i.Here xξ stands for the scalar product in R n .Both F and We define the sequences ϕ = {ϕ j } j∈N 0 and ϕ = {ϕ j } j∈Z form a dyadic resolution of unity, respectively.The entire analytic functions (ϕ j f ) ∨ (x) (j ∈ N 0 ) and (ϕ j f ) ∨ (x) (j ∈ Z) make sense pointwise in R n for any f ∈ S ′ (R n ).
We are interested in inhomogeneous Besov and Triebel-Lizorkin spaces A s p,q (R n ) with A ∈ {B, F } with s ∈ R and 0 < p, q ≤ ∞.The standard norms of these spaces and their homogeneous counterparts are given as follows is finite (with the usual modification if q = ∞).
Remark 2.2.We recall that all spaces defined above are independent of the respective resolution of unity ϕ according to (2.3)-(2.6)(equivalent quasi-norms).This justifies the omission of the subscript ϕ in (2.7)-(2.10) in the sequel (and any other marks in connection with equivalent quasi-norms).Note that the spaces A s p,q (R n ) are translation invariant.This follows easily from elementary properties of the Fourier transform and the translation invariance of L p -spaces.The theory of inhomogeneous spaces, including special cases and their history may be found in [14], [15], [16] and [20].As far as homogeneous spaces are concerned we refer to [14,Chapter 5] as well as to [19,Definition 2.8] as far as (2.10) is concerned.We will use these spaces only in the context of norm equivalences.Especially for our purposes it is not necessary to discuss the usual ambiguity of homogeneous spaces.Finally, We need a few specific properties of the above defined spaces.Let ϕ 0 and ϕ = {ϕ j } j∈Z be as in (2.3) and (2.5).Let 1 ≤ p, q ≤ ∞ (p < ∞ for F -spaces) and s > 0. Then are equivalent norms in B s p,q (R n ) and are equivalent norms in F s p,q (R n ) (with the usual modification if q = ∞).This is a special case of corresponding assertions in [15,Section 2.3.3,where one finds also continuous versions with t > 0 in place of 2 −j , j ∈ Z, which are nearer to what follows.These norms are characterizing what means that f ∈ S ′ (R n ) belongs to A s p,q (R n ) if, and only if, the corresponding norm is finite.We need an extension of the above norms to a wider class of functions ϕ j (x) = ϕ 0 (2 −j x) and ϕ 0 (tx). and and and Proof.The extension of (2.11), (2.12) from ϕ 0 (2 −j x) and its continuous counterpart ϕ 0 (tx) to the above assertion follows from [19, Proposition 2.10, pp.[18][19] and the references given there specified to the above values of the parameters s, p, q.Again these norms are characterizations as explained above.
Remark 2.4.Note that by [19,Proposition 2.10] the second summands on righthand-sides in (2.17) and (2.20) are equivalent norms in Ḃs p,q (R n ) and Ḟ s p,q (R n ), respectively, for all admitted parameters.
Let us shortly comment on the conditions with respect to ϕ 0 and ϕ supposed in Proposition 2.3.The system (ϕ j ) ∞ j=1 introduced in order to define the spaces A s p,q (R n ) (see Definition 2.1) can be rewritten as and ϕ 0 has the meaning of (2.3).The function ̺ has compact support in {x : 1 2 ≤ |x| ≤ 3  2 } and satisfies the Tauberian condition |̺(x)|> 0 on {x : 3 4 ≤ |x| ≤ 1}.The characterization of spaces A s p,q (R n ) in Proposition 2.3 can be considered as a continuous extension and generalization of Definition 2.1, where the generating function ̺ is replaced by ϕ.In contrast to the properties of ̺ it is not assumed that ϕ has compact support in a subset of R n \ {0}.Conditions (2.15) and (2.18) ensure sufficiently strong decay to 0 near the origin, whereas (2.16)

Characterizations of some function spaces in terms of fractional Gauss-Weierstrass semi-groups
We wish to apply Proposition 2.3 to is not smooth at ξ = 0 and some extra care is needed.This is just the reason why we prefer now Proposition 2.3 (under the indicated restrictions for the underlying spaces) compared with the original inhomogeneous versions according to [15, Theorems 2.4.1, 2.5.1, pp.100, 101, 132] (which apply to all spaces A s p,q (R n ) with exception of F s ∞,q (R n )).Rescue comes from the following observations in [12].Let and according to (1.6) (2.23) Then the estimates (2.26) In the distinguished case α = 1 one has now final characterizations of all spaces A s p,q (R n ) with s ∈ R and 0 < p, q ≤ ∞ in terms of ∂ k t W t w for the classical Gauss-Weierstrass semi-group W t = W 1 t .This may be found in [20, Section 3.2.7,pp.106-109] and the references given there.We extend now these assertions to the fractional Gauss-Weierstrass semi-group W α t under the same restrictions for the spaces A s p,q (R n ) as in Proposition 2.3.Theorem 2.5.Let ϕ 0 be as in (2.3) and W α t be as in Proof.
Step 1.We rely on part (i) of Proposition 2.3 choosing there (2.29) Then (2.15) follows from (2.24) and Secondly we have to justify (2.16) with ϕ as in (2.29).But this follows from n/2 < l ∈ N, where W l 2 (R n ) are the classical Sobolev spaces.Then the second terms in (2.17) are equivalent to again with ϕ as in (2.29) and τ = t . One has by ϕ t and (1.2) that ϕ(t Inserted in (2.32) one obtains (2.27).
Step 2. For the proof of part (ii) we rely on part (ii) of Proposition 2.3 choosing such that the corresponding norm is finite.In particular it follows from the above considerations immediately that always But we will not stress this point in the sequel.

Smoothing properties
We justify (1.1)-(1.3).As already mentioned in the Introduction assertions of this type are not new.A proof of (1.1) with α = 1 for the classical Gauss-Weierstrass semi-group W t w = W 1 t w covering all spaces A s p,q (R n ), A ∈ {B, F }, s ∈ R and 0 < p, q ≤ ∞ may be found in [20,Theorem 3.35,p. 110].It relies on characterization of these spaces in terms of W t w using in a decisive way that the underlying kernel e −|ξ| 2 ∈ S(R n ) is smooth at the origin ξ = 0.This is no longer the case in general if one steps from W t to W α t , α > 0, this means from e −|ξ| 2 to e −|ξ| 2α .On the other hand, (1.1) for the classical Gauss-Weierstrass semi-group W t = W 1 t , restricted to A ∈ {B, F }, s ∈ R and 1 ≤ p, q ≤ ∞ (p < ∞ for F -spaces) is also a special case of a corresponding assertion for related hybrid spaces L r A s p,q (R n ).This may be found in [18, Theorem 4.1, p. 114] including related references and comments.The extension of (1.1) from α = 1 to α ∈ N for the spaces A ∈ {B, F }, s ∈ R and 1 ≤ p, q ≤ ∞ (p < ∞ for F -spaces) goes back to [2, Theorem 3.5, p. 2123].The arguments both in [18] (including underlying references) and [2] rely on the elaborated machinery of (caloric) wavelet expansions.The step from α ∈ N to α > 0 in (1.1) for A ∈ {B, F }, s ∈ R and 1 ≤ p, q ≤ ∞ is covered by the recent paper [9] in the larger context of convolution inequalities in these spaces.What follows may be considered as a surprising simple proof of these assertions relying on Theorem 2.5 and a few well-known properties of the spaces A s p,q (R n ) as introduced in Definition 2.1.
Theorem 2.7.Let W α t be as in (1.2).Let A ∈ {B, F }, s ∈ R and 1 ≤ p, q ≤ ∞.Let d ≥ 0. Then there is a constant c > 0 such that for all t with 0 < t ≤ 1 and all w ∈ A s p,q (R n ), (2.37) for the second summand on the right-hand side of (2.27).According to (1.2) and (2.24) we have f = W α τ w ∈ L p (R n ) (see also Remark 2.6) and it holds (2.39) (2.40) Note that (2.39) is well defined due to (2.24).Combining (2.38) and (2.40) one obtains As far as the first terms on the right-hand side of (2.27) are concerned it is sufficient to justify for some c > 0 and all 0 < τ ≤ 1. Recall that 1 ≤ p ≤ ∞.Then (2.45) follows from for some C > 0 and all 0 < λ < ∞ what in turn can be obtained from (2.22), (2.24) and (2.47) Now (2.37) can be obtained for B s p,q (R n ) with s > 0 and 1 ≤ p, q ≤ ∞ from (2.27), (2.44) and (2.45).Next we consider the case of F -spaces.Let s > 0 and let ω ∈ Again it holds (2.40).The counterpart of (2.41) reads as where d ≥ 0 and s+d 2α + 1 q < k ∈ N. By the same arguments as in the proof of (2.44) one obtains (2.50) Now (2.37) for A = F is a consequence of (2.28), (2.45) and (2.50).
Step 2. Recall A ∈ {B, F }, where s > 0 and σ > 0 (see Remark 2.8 below).If s ≤ 0 and f ∈ A s p,q (R n ) then we take (2.53) as definition of W α t f , where σ is chosen such that s + σ > 0. Then one can extend (2.37) from the spaces A s p,q (R n ), s > 0 and 1 ≤ p, q ≤ ∞ (p < ∞ for F -spaces) treated in Step 1 to their counterparts with s ≤ 0. This covers all spaces in the above theorem with exception of for the related dual spaces in the framework of the dual pairing S(R n ), S ′ (R n ) .This is a special case of [20, (1.25), p. 5] with a reference to [10, Theorem 4, p. 87] as far as the case q = ∞ is concerned (if q < ∞ then S(R n ) is already dense in F s 1,q (R n ) and the related duality is well known, [20, p. 5] and the references there).If ϕ ∈ S(R n ) then W α t ϕ can be approximated in, say, F s+1 1,1 (R n ) by functions belonging to S(R n ) for any s.But then it follows by embedding that this is also an approximation in F s 1,∞ (R n ) for any s.In particular one has by Step 2 (2.55) The operator

54), (2.55).
Remark 2.8.We justify (2.53).Let f ∈ A s p,q (R n ), 1 ≤ p < ∞, s > 0 and σ > 0. Without loss of generality we may assume σ < ∞ (otherwise one replaces s by is well defined pointwise and belongs to S ′ (R n ).Hence, for all f ∈ S(R n ).Then it follows (2.53) for all f ∈ A s p,q (R n ) by (2.37) (with d = 0), the lift property (2.37) and density of S(R n ) in A s p,q (R n ).
Remark 2.9.We observe that f | * A s p,q (R n ) is an equivalent norm in Ȧs p,q (R n ) if 2αk > s for Besov spaces and 2αk > s + n for Triebel-Lizorkin spaces.This is a direct consequence of Remark 2.2 and Theorem 2.5.

Nonlinear fractional heat equations
In [3] we dealt with the Cauchy problem (1.4), (1.5) where α > 0 is a natural number.The case α = 1 corresponds to a classical non-linear heat equation.We established mild and strong solutions in appropriate function spaces The aim of this section is to extend some of these results to the case of fractional powers α > 1/2.This restriction results from the mapping properties of the non-linearity Du 2 in A s p,q -spaces, see Proposition 3.3 below.In particular, we make use of the smoothing properties formulated in Theorem 2.7.
For later purposes we recall multiplication properties in the respective spaces A s p,q (R n ) derived in [3] including p = ∞ for F -spaces.
for all f, g ∈ A s p,q (R n ) and all ε > 0. Proof.
for all f, g ∈ A s p,q (R n ).
Next we derive an estimate of T u 0 as defined in (1.7) for fixed t > 0 in appropriate function spaces A s p,q (R n ).
< s < ∞ and let α > 1/2.Let T > 0 and let a, v, d such that then there exists a constant c > 0, independent of u 0 and u, such that for all t with 0 < t < T (with 1 v = 0 and the modification (1.9) if v = ∞).
Proof.Note that condition (3.3) with respect to d implies α > 1 2 .Using Theorem 2.7 and Corollary 3.2 with s − d in place of σ we can estimate as follows Here we used that according to (3.3).By means of Hölder's inequality with exponent αv > 1 we obtain Here we used the conditions a + 1 v < α as well as d < 2(α − 1 v ) to ensure that the integral is finite.
Let u 0 ∈ A s 0 p,q (R n ).(i) Then there exists a number T > 0 such that the Cauchy problem (1.4),(1.5)has a unique mild solution u belonging to for all s satisfying s 0 ≤ s < s 0 + min(α, 2α − 1) (3.10) where a, v such that (ii) The mild solution u obtained in part (i) also belongs to the space L ∞ ((0, T ), A s 0 p,q (R n )).Moreover, if, in addition, max(p, q) < ∞ then the above solution u(•) converges to u 0 with respect to the norm in Proof.First we observe that assumptions (3.8) and (3.10) imply that Hence, conditions (3.12) and (3.13) make sense.
Step 1.We choose d such that which is possible due to conditions (3.12) and (3.13).Then it follows from Proposition 3.3 that for 0 < t < T .We multiply both sides with t a 2α .Raising to the power of 2αv and integrating over (0, T ) yield and Thus, T u 0 maps the unit ball U T in L 2αv ((0, T ), a 2α , A s p,q (R n )) into itself if T is sufficiently small.As for the contraction property consider u, v ∈ U T .A similar calculation with d as above (cf.also (3.6) and (3.7)) yields Application of Proposition 3.1 leads in combination with Hölder's inequality to Let temporarily X s T = L 2αv ((0, T ), a 2α , A s p,q (R n )), then it follows from (3.17) with the same κ as in (3.16).If T > 0 is small enough, then T u 0 : U T → U T is a contraction.Since we deal with Banach spaces we have shown that T u has a unique fixed point in U T and hence a mild solution of the Cauchy problem (1.4), (1.5).
To extend the uniqueness to the whole space we proceed similarly to e.g.[11], [21].
Let u ∈ U T be the above solution and v ∈ X s T a second solution.We observe that (3.18) holds for any 0 < t ≤ T 0 ≤ T .With u ∈ U T we obtain If we choose T 0 > 0 small enough such that c T κ 0 (1 + v|X s T ) < 1 it follows that u(•, t) = v(•, t) for any t ∈ (0, T 0 ].Now we take u(•, T 0 ) ∈ A s p,q (R n ) ֒→ A s 0 p,q as new initial value and proceed as in the previous steps until (3.19) inclusively.There exists a unique solution u in a neighbourhood U δ (T 0 ) with u(•, T 0 ) = u(•, T 0 ).Since it holds that u(•, t) = u(•, t) for all t ∈ (0, T 0 ] ∩ U δ (T 0 ) we have extended u to some interval (0, T 1 ] with T 0 < T 1 .Thus, we have prolongated u(•, t) − v(•, t) = 0 to some interval (0, T 1 ] where T 0 < T 1 ≤ T By iteration it follows the uniqueness in L 2αv ((0, T ), a 2α , A s p,q (R n )).
Step 2. We show part (ii) of the theorem.To this end we first prove that the mild solution obtained in Step 1 belongs to L ∞ ((0, T ), A s 0 p,q (R n )).Let u 0 ∈ A s 0 p,q (R n ) and let u ∈ L ∞ (0, T ), a 2α , A s p,q (R n ) be the corresponding solution, where s and a satisfy (3.10), (3.11) and (3.13),where 1  v is replaced by 0. Let 0 < t < T .It holds Taking into account (2.53) and the lift property (2.51) we may assume s 0 > 0.
Concerning the first summand we obtain independent of t, where (3.20) follows from the generalized Minkowski inequality for Banach spaces and (3.21) from the translation invariance of A s p,q (R n )-spaces (see also Remark 2.2) and (2.24).In order to estimate the second summand we first consider the case that s − s 0 ≤ 1 + n p − s + and we put where ε is chosen such that 0 < ε < 2α − 1 − ( n p − s + − (s − s 0 ) according to (3.13).Then, d > 0, s − s 0 < α − d 2 , and we may choose a such that Applying Theorem 2.7 with s 0 −d in place of s and Corollary 3.
The boundedness of u(t, •) on (0, T ) in A s 0 p,q (R n ) follows from (3.21), (3.22) (due to a < α − d 2 ), and (3.23) (because of a < α).Next we consider the limit of u(t, The second summand on the right-hand side tends to zero if t → 0+ as a consequence of (3.22), (3.23) and the conditions with respect to a, α, and d.Using the identity we obtain the estimate The first summand is lower than ε if we choose N large enough.Fixing this N the second summand tends to zero for t tending to zero.This follows from the fact that the Schwartz space S(R n ) is dense in A s p,q (R n ) if max(p, q) < ∞ and the continuity of the translation (See also [5,Subsection 1.2.d] for more details with respect to approximate identities).This completes the proof.
In addition to the results of the previous part one may ask for well-posedness of the Cauchy problem.The notation well-posedness is not totally fixed in the literature (see the comments in [17, Subsection 6.2.5])We adapt the standard notation, see e.g.[4].The Cauchy problem is called locally well-posed if there exists a unique mild and strong solution according to Theorem 3.4.In addition it is required continuous dependence of the solutions with respect to initial data.This means that for solutions u 1 and u 2 of (1.4), (1.5) according to Theorem 3.4 with respect to initial data u 1 0 and u 2 0 , respectively, for any ε > 0 there exists a δ > 0 and a time T > 0 such that for all 0 < t < T u Recall that by construction of the solution as a fixed point of T u 0 we may assume u|L 2) be solutions of (1.4), (1.5) obtained in Theorem 3.4 with initial data u i 0 ∈ A s 0 p,q (R n ) in the corresponding time interval (0, T i ).Let max(p, q) < ∞.Then under the conditions of Theorem 3.4 the Cauchy problem (1.4), (1.5) is locally well-posed.
Proof.Let u 1 , u 2 be two solutions of (1.4), (1.5) with corresponding initial data u 1 0 , u 2 0 .We have To estimate the first summand of the right-hand side we use again Theorem 2.7 with d = 0.The second summand can be treated in the same way as in Step 2 of the proof of Theorem 3.4 with . Hence, application of Minkowski's inequality leads to with the same choice of d ≥ 0 as in step 2 of Theorem 3.4 and for all 0 < t < T with T ≤ min(T 1 , T 2 ).Then the right hand side in (3.29) is lower the the given ε if this T is chosen small enough.(ii) If s 0 > n p − 2α + 1 + and 1 ≤ p ≤ ∞ then there exists a unique mild solution u ∈ L 2αv (0, T ), a 2α , A s p, q (R n ) , where s 0 ≤ s < s 0 + α.Supercritical spaces A s 0 p,q (R n ) are completely covered if 1 ≤ p ≤ n α−1 (see Figure 3).Initial data u 0 ∈ A 0 p,q (R n ) are admitted provided that n 2α−1 < p ≤ ∞ if Remark 4.4.The case α > n 2 + 1.In this case (3.9) implies (3.8).We have and 1 ≤ p ≤ ∞ then there exists a unique mild solution u ∈ L 2αv (0, T ), a 2α , A s p, q (R n ) , where s 0 ≤ s < s 0 + α.Supercritical spaces A s 0 p,q (R n ) can never be completely covered for given p, 1 ≤ p ≤ ∞ (see Figure 4).Initial data u 0 ∈ A 0 p,q (R n ) are admitted for all p, 1 ≤ p ≤ ∞.Remark 4.5.Some attention attracted Cauchy problems of type (1.4), (1.5) for fractional power dissipative equations with initial data belonging to spaces L p (R n ), 1 < p < ∞ (see, for example, [12]).Let us suppose that initial data u 0 ∈ A 0 p,q (R n ), where 1 < p < ∞, 1 ≤ q ≤ ∞.In particular, this applies to u 0 ∈ L p (R n ) = F 0 p,2 (R n ).We follow our approach and recall (see the preceding remarks) that u 0 ∈ A 0 p,q (R n ) is admitted if 0 < 1 p < 2α−1 n (0 < 1 p ≤ 1 if 2α−1 n > 1) for given α > In the following we shall make use of the embeddings for λ ≥ 0 and A n p − n r p,q (R n ) ֒→ A 0 r,q (R n ) (4.3) for 0 < 1 r < 1 p .We always assume u 0 ∈ A 0 p,q (R n ) and u stands for the unique mild solution according to Theorem 3.4.We distinguish the following cases:
from the smoothing property (1.1) a key ingredient in the proof turns out to be the mapping property of the nonlinearity Du 2 = n j=0 ∂ j u 2 in (1.4) considered in Proposition 3.1 and Corollary 3.2 which require the condition s > n p − n 2 + < d < 2α and a < α.If s − s 0 > 1 + n p − s + then we can apply Theorem 2.7 with d = 0 and Corollary 3.2 with s 0 in place of σ to get t 0

Figure 4 :
Figure 4: α > n 2 + 1 Let us also mention that the condition with respect to ϕ 0 ∈ C ∞ 0 (R n ) can be weakened.For a more detailed discussion we refer to[15, Corollary 2.4.1/1,Remark 2.4.1/3].Relevant examples will be treated in the next subsection.