1. A Heat-Wave equation in Morrey spaces
Throughout this notes, the symbol stands for the set of all natural numbers. For , we always let be -dimensional Euclidean space with Euclidean norm denoted by and endowed with Lebesgue measure . Also stands for -dimensional upper half-space with Lebesgue measure . Let be a vector field on upper half-space , we denote . Here is so-called absolute value of .
Here we are interested in a semilinear integro-partial differential equation, which interpolates the semilinear heat and wave equations which reads as follows
where denotes the Gamma function, is the Laplacian in the -variable and is a fluid at time and position that assumes the given data (initial velocity) . This equation is formally equivalent to the initial value problem for the time-fractional partial differential equation (FPDE)
where and stands for the Riemann-Liouville derivative of order given at (16).
Differential equations of time-fractional order appear naturally in several fields such as physics, chemistry and engineering by modelling phenomena in viscoelasticity, thermoelasticity, processes in media with fractal geometry, heat flow in material with memory and many others. The two most common types of fractional derivatives acting on time variable are those of Riemann-Liouville and Caputo. We refer the surveys Kilbas and Povstenko in which the reader can find a good bibliography for applications on those fields. Models with fractional derivatives can naturally connect structurally different groups of PDEs and their mathematical analysis may give information about the transition (or loss) of basic properties from one to another. Two groups are the parabolic and hyperbolic PDEs whose well-posedness and asymptotic behavior theory presents a lot of differences. For instance, in weak- Besov-spaces, Morrey spaces and other ones, there is an extensive bibliography for global well-posedness and asymptotic behavior for semilinear heat equations (and other parabolic equations). On the other hand, for semilinear wave equations, although there exist results in , weak and Besov spaces, there is no results in Morrey spaces. The main reason is the loss of decay of the semigroup (and its time-derivative) associated to the free wave equation, namely and So, it is natural to wonder what would be the behavior of the semilinear (FPDE) in the framework of Morrey spaces, which presents a mixed parabolic-hyperbolic structure.
Between interesting points obtained, let us comment about technical ones. Due to the semigroup property (21), further restrictions appear in our main theorems in comparison with classical nonlinear heat equations. Making the derivative index go from to the estimates and corresponding restrictions become worse, and they are completely lost when reaches the endpoint (see Lemmas 4, 5). The proof of the pointwise estimate (23) shows that the worst parcel in (21) is the term (see (21)). Then In particular, notice that for (heat semigroup) the upper bound on parameter is not necessary, that is, one can take Finally, based on above observations, our results and estimates suggest the following: the semilinear wave equation () in is not well-posed in Morrey spaces. The mathematical verification of this assertion seems to be an interesting open problem.
2. Morrey spaces
In this section some basic properties about Morrey spaces are reviewed. For further details on theses spaces, the reader is referred to Kato, Peetre, Taylor. Let denote the open ball in centered in the origin and with radius . For two parameters and , we define the Morrey spaces as the set of functions such that
where denotes a constant independent of and . The space endowed with the norm
is a Banach space. For and the homogeneous Sobolev-Morrey space is Banach with norm
Taking , stands for the total variation of on and is a space of signed measures In particular, when is the space of finite measures. For and is the homogeneous Sobolev space With the natural adaptation in (3) for the space corresponds to .
Morrey spaces present the following scaling
where the exponent is called scaling index. We have that
Let us define the closed subspace of (denoted by ) by means of the property if and only if
This subspace is useful to deal with semigroup of convolution operators when the respective kernels present a suitable polynomial decay at infinity. In general, such semigroups are only weakly continuous at in but they are -semigroups in as it is the case of . This property is important in order to derive local-in-time well-posedness for PDEs.
Morrey spaces contain Lebesgue and Marcinkiewicz spaces with the same scaling indexes. Precisely, we have the continuous proper inclusions
Lemma 1 Suppose that , and .
- (Inclusion) is decreasing in , i.e., if and then
- (Sobolev type embedding) If , and then
- (Holder inequality) If and then and
- (Homogeneous function) Let , and . Then , for all .
Lemma 2 Let and and Assume that there is such that
for all with and for all Then the multiplier operator on is bounded from to and the following estimate hold true
where is the set of equivalence classes in modulo polynomials with variables.
3. Mittag-Leffler Function
Differential equations of time-fractional order is old, but even being old, the fractional calculus is little studied by mathematicians. Possibly, because many of them are unfamiliar with this topic and its applications in various sciences. After Liouville and Riemann, deep developments was obtained by many authors. Today, time-fractional order integral and derivative is known as Riemann-Liouville’s integral and derivative. More precisely, let be a Lebesgue integrable function in and , Riemann-Liouville’s integral is defined by
and Riemann-Liouville’s derivative by
where , , being denoted by so-called Gamma function.
Hardy and Littlewood give wider properties about this integrals. His showed, for instance, the boundedness of from to , for . This result, is well known as Hardy-Littlewood-Sobolev theorem, Sobolev because of its importance in the theory of fractional Sobolev Spaces.
Related the theory of partial differential equations, much works is devoted to find an unified theory of Green functions associated to fractional problems. In this way, fractional diffusion-wave equation is a celebrity, and an unified theory of the heat equation and wave equation was obtained. Fujita, Schneider and Wyss, for instance, founds the special function to represent the Green function associated to the linear part of the diffusion-wave problem (FPDE)
such special function, is knew as Mittag-Leffler function. More precisely,
and the Green function, defined via Fourier inversion, is given by
which generate the following semigroup of operators ,
In what follows, we recall some functions which is useful to handle the symbol of the semigroup (see (19)). For let us set
Lemma 3 Let and be as in (18). We have that
for all Moreover, , is a probability measure, and
The desired identity follows by taking in the last equality.
4. Some estimates
The aim of this section is to derive estimates for the semigroup For that matter we will need pointwise estimates for the fundamental solution in Fourier variables.
Lemma 4 Let and There is such that
for all with and for all
In the sequel we prove key estimates on Morrey spaces for the semigroup
Lemma 5 Let , , and . There exists such that
Proof: Let and defined through Consider the multiplier operators
because of (26).
5. Self-similarity and symmetries for a fractional-wave equation
which should be understood in the Bochner sense in Morrey spaces, being . Throughout this notes a mild solution for (1) is a function satisfying (27). We shall employ the Kato-Fujita method (see [Kato]) to integral equation to get our results.
From now on, we perform a scaling analysis in order to choose the correct indexes for Kato-Fujita spaces. A simple computation by using
shows that the indexes and are the unique ones such that the function given by
is a solution of that, for each whenever is also. The scaling map for (29) is defined by
Making in (31) one obtains the following scaling for the initial condition
Solutions invariant by (31), that is
Consider the parameters
and let stands for the class of bounded and continuous functions from to a Banach space We take belonging to the critical space and study (27) in the Kato-Fujita type space
which is Banach with the norm
From Lemma 1, a typical data belonging to is the homogeneous function
where and is a bounded function on sphere We refer the book [Gigabook, chapter 3] for more details about self-similar solutions and PDE’s.
Our well-posedness result reads as follows.
Theorem 6 (Well-posedness) Let , , , and . Suppose that and
- (i) (Existence and uniqueness) There exist and such that if then the equation (1) has a mild solution , which is the unique one in the ball . Moreover, in as
- (ii) (Continuous dependence on data) Let The data-solution map is Lipschitz continuous from to .
- (i) With a slight adaptation of the proof of Theorem 6, we could treat more general nonlinearities. Precisely, one could consider (1) and (29) with instead of , where , and there is such that
- (ii) (Local-in-time well-posedness) A local version of Theorem 6 holds true by replacing the smallness condition on initial data by a smallness one on existence time . Here we should consider the local-in-time spaceand such that In particular, this condition is verified when belongs to the closed subspace (see (8)).
Let be the orthogonal matrix group in and let be a subset of A function is said symmetric and antisymmetric under the action of when and , respectively, for every .
Theorem 7 Under the hypotheses of Theorem 6.
- (i) (Self-similarity) If is a homogeneous function of degree , then the mild solution given in Theorem 6 is self-similar.
- (ii) (Symmetry and antisymmetry) The solution is antisymmetric (resp. symmetric) for , when is antisymmetric (resp. symmetric) under
- (iii) (Positivity) If and (resp. ) then is positive (resp. negative).
Remark 2 (Special examples of symmetry and antisymmetry)
- (i) The case corresponds to radial symmetry. Therefore, it follows from Theorem 7 (ii) that if is radially symmetric then is radially symmetric for .
- (ii) Let be the reflection over the origin and let be the identity map. The case corresponds to parity of functions, that is, is even and odd when and , respectively. So, from Theorem 7 (ii), we have that the solution is even (resp. odd) for when is even (resp. odd).