Heat-wave equation in Morrey spaces


1. A Heat-Wave equation in Morrey spaces 

This notes is an exposition of the article written by me in join with full professor Lucas Catão de Freitas Ferreira and therefore no proofs will be given in general.

Throughout this notes, the symbol {\mathbb{N}} stands for the set of all natural numbers. For {n\in\mathbb{N}}, we always let {\mathbb{R}^{n}} be {n}-dimensional Euclidean space with Euclidean norm denoted by {\Vert x\Vert} and endowed with Lebesgue measure {dx}. Also {\mathbb{R}^{n+1}_{+}} stands for {n+1}-dimensional upper half-space {\{(x,t)\,:\, x\in\mathbb{R}^{n},\, t>0\}} with Lebesgue measure {dxdt}. Let {u:\mathbb{R} ^{n+1}_{+}\rightarrow\mathbb{R}^{n}} be a vector field on upper half-space {\mathbb{R}^{n+1}_{+}}, we denote {\vert u\vert=\max_{i=1,...,n}\vert u_{i}(x,t) \vert_{\mathbb{R}}}. Here {\vert u_{i}(x,t)\vert_{\mathbb{R}}} is so-called absolute value of {u_{i}(x,t)\in\mathbb{R}}.

Here we are interested in a semilinear integro-partial differential equation, which interpolates the semilinear heat and wave equations which reads as follows

\displaystyle u(x,t)=u_{0}(x)+\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1} (\Delta_{x}u(x,s)+|u(x,s)|^{\rho}u(x,s))ds \ \ \ \ \ (1)

where {1<\alpha<2,} {0<\rho<\infty,} {\Gamma(\alpha)} denotes the Gamma function, {\Delta_{x}} is the Laplacian in the {x}-variable and {u=u(x,t)=(u_{1}(x,t),\cdots,u_{n}(x,t))} is a fluid at time {t\in [0,\infty)} and position {x\in\mathbb{R}^{n}} that assumes the given data (initial velocity) {u_{0}=u_{0}(x)}. This equation is formally equivalent to the initial value problem for the time-fractional partial differential equation (FPDE)

\displaystyle \begin{array}{rcl} \mathbf{\partial}_{t}^{\alpha}u =\Delta_{x}u+|u|^{\rho}u,\text{ in }\mathbb{R}^{n},\;t>0,\\ u(x,0)=u_{0}\text{ and }\partial_{t}u(x,0)=0\text{ in }\mathbb{R}^{n}, \end{array}

where {\mathbf{\partial}_{t}^{\alpha}u=\mathbf{D}_{t}^{\alpha-1}(\partial _{t}u)} and {\mathbf{D}_{t}^{\alpha-1}} stands for the Riemann-Liouville derivative of order {\alpha-1} 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 {t} 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 {L^{p},} weak-{L^{p},} 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 {L^{p}}, weak{-L^{p}} 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 {(\frac {\sin(\left\vert \xi\right\vert t)}{\left\vert \xi\right\vert t})^{\vee}} and {(\cos(\left\vert \xi\right\vert t))^{\vee}.} 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 {\alpha} go from {\alpha=1} to {\alpha=2,} the estimates and corresponding restrictions become worse, and they are completely lost when {\alpha} reaches the endpoint {2} (see Lemmas 45). The proof of the pointwise estimate (23) shows that the worst parcel in (21) is the term {l_{\alpha}(\xi)} (see (21)). Then In particular, notice that for {\alpha=1} (heat semigroup) the upper bound on parameter {\delta} is not necessary, that is, one can take {\delta\in\lbrack0,\infty).} Finally, based on above observations, our results and estimates suggest the following: the semilinear wave equation ({\alpha=2}) in {\mathbb{R}^{n}} 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 KatoPeetreTaylor. Let {\mathbb{D}_{r}} denote the open ball in {\mathbb{R}^{n}} centered in the origin and with radius {r>0}. For two parameters {1\leq p<\infty} and {0\leq\mu<n}, we define the Morrey spaces {\mathcal{M}_{p,\mu}=\mathcal{M} _{p,\mu}(\mathbb{R}^{n})} as the set of functions {f\in L^{p}(\mathbb{D}_{r})} such that

\displaystyle \Vert f\Vert_{L^{p}(\mathbb{D}_{r})}\leq C\,r^{\frac{\mu}{p}}, \ \ \ \ \ (2)

where {C>0} denotes a constant independent of {x_{0},r} and {f}. The space {\mathcal{M}_{p,\mu}} endowed with the norm

\displaystyle \Vert f\Vert_{p,\mu}=\sup_{\mathbb{D}_{r}}r^{-\frac{\mu}{p}}\Vert f\Vert_{L^{p}(\mathbb{D}_{r})} \ \ \ \ \ (3)

is a Banach space. For {s\in\mathbb{R}} and {1\leq p<\infty,} the homogeneous Sobolev-Morrey space {\mathcal{M}_{p,\mu}^{s}=(-\Delta)^{-s/2}\mathcal{M} _{p,\mu}} is Banach with norm

\displaystyle \left\Vert f\right\Vert _{\mathcal{M}_{p,\mu}^{s}}=\Vert (-\Delta )^{s/2}f\Vert _{p,\mu}. \ \ \ \ \ (4)

Taking {p=1}, {\Vert f\Vert_{L^{1}(\mathbb{D}_{r})}} stands for the total variation of {f} on {\mathbb{D}_{r}} and {\mathcal{M}_{1,\mu}} is a space of signed measures{.} In particular, when {\mu=0,} {\mathcal{M}_{1,0}} {=\mathcal{M}} is the space of finite measures. For {p>1,} {\mathcal{M} _{p,0}=L^{p}} and {\mathcal{M}_{p,0}^{s}=\dot{H}_{p}^{s}} is the homogeneous Sobolev space{.} With the natural adaptation in (3) for {p=\infty,} the space {L^{\infty}} corresponds to {\mathcal{M}_{\infty,\mu}}.

Morrey spaces present the following scaling

\displaystyle \Vert f(\lambda x)\Vert_{p,\mu}=\lambda^{-\frac{n-\mu}{p}}\Vert f\Vert_{p,\mu} \ \ \ \ \ (5)

and

\displaystyle \left\Vert f(\lambda x)\right\Vert _{\mathcal{M}_{p,\mu}^{s}}=\lambda ^{s-\frac{n-\mu}{p}}\left\Vert f(x)\right\Vert _{\mathcal{M}_{p,\mu}^{s} }\text{,} \ \ \ \ \ (6)

where the exponent {s-\frac{n-\mu}{p}} is called scaling index. We have that

\displaystyle (-\Delta)^{l/2}\mathcal{M}_{p,\mu}^{s}=\mathcal{M}_{p,\mu}^{s-l}. \ \ \ \ \ (7)

Let us define the closed subspace of {\mathcal{M}_{p,\mu}} (denoted by {\ddot{\mathcal{M}}_{p,\mu}}) by means of the property {f\in\ddot{\mathcal{M} }_{p,\mu}} if and only if

\displaystyle \Vert f(\cdot-y)-f(\cdot)\Vert_{p,\mu}\rightarrow0\text{ as }y\rightarrow0. \ \ \ \ \ (8)

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 {t=0^{+}} in {\mathcal{M}_{p,\mu},} but they are {C_{0}}-semigroups in {\ddot{\mathcal{M} }_{p,\mu},} as it is the case of {\{G_{\alpha}(t)\}_{t\geq0}}. 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

\displaystyle L^{q}(\mathbb{R}^{n})\varsubsetneq L^{q,\infty}(\mathbb{R}^{n})\varsubsetneq \mathcal{M}_{p,\mu}(\mathbb{R}^{n}) \ \ \ \ \ (9)

where {p<q} and {\mu=n(q-p)/q} (see e.g. Miyakawa or MF de Almeida and LFC Ferreira).

In the next lemma, we remember some important inequalities and inclusions in Morrey spaces, see e.g. KatoTaylor.

Lemma 1 Suppose that {s_{1},s_{2}\in\mathbb{R}}, {1\leq p,q,r<\infty} and {0\leq\lambda,\mu,\upsilon<n}.

  • (Inclusion) {\mathcal{M}_{p,\mu}} is decreasing in {p}, i.e., if {p\leq q} and {\frac{n-\mu}{p} =\frac{n-\lambda}{q}} then

    \displaystyle \mathcal{M}_{p,\mu}\supseteq\mathcal{M}_{q,\lambda} \ \ \ \ \ (10)

  • (Sobolev type embedding) If {p\leq q}, {s_{2}\geq s_{1}} and {s_{2}-\frac{n-\mu}{p}=s_{1}-\frac{n-\mu}{q}} then

\displaystyle \mathcal{M}_{p,\mu}^{s_{2}}\subset\mathcal{M}_{q,\mu}^{s_{1}} \ \ \ \ \ (11)

  • (Holder inequality) If {\;\frac{1}{r}=\frac{1} {p}+\frac{1}{q}} and {\frac{\upsilon}{r}=\frac{\lambda}{p}+\frac{\mu}{q}} then {fg\in\mathcal{M}_{r,\upsilon}} and   

\displaystyle\Vert fg\Vert_{r,\upsilon}\leqslant\Vert f\Vert_{p,\lambda}\Vert g\Vert _{q,\mu}. \ \ \ \ \ (12)

  • (Homogeneous function) Let {\Omega\in L^{\infty }(\mathbb{S}^{n-1})}, {0<d<n} and {1\leq r<n/d}. Then {\Omega(x/|x|)|x|^{-d} \in\mathcal{M}_{r,n-dr}}, for all {x\in{\mathbb{R}^{n}\backslash\{0\}}}.

We finish this section by recalling estimates for certain multiplier operators on {\mathcal{M}_{p,\mu}^{s}} see e.g. Kozono-YamazakiKozono-YamazakiTaylor for lemma below.

In the next posts, inspired in the work of Taylor, we given a proof based in the Theorem 7 given in the post A fractional Hörmander type multiplier theorem.

Lemma 2 Let {m,s\in\mathbb{R},} {1<p<\infty} and {0\leq\mu<n} and {F(\xi)\in C^{[n/2]+1}(\mathbb{R}^{n}\backslash\{0\}).} Assume that there is {A>0} such that

\displaystyle \left\vert \frac{\partial^{k}F}{\partial\xi^{k}}(\xi)\right\vert \leq A\left\vert \xi\right\vert ^{m-\left\vert k\right\vert },\text{ } \ \ \ \ \ (13)

for all {k\in(\mathbb{N}\cup\{0\})^{n}} with {\left\vert k\right\vert \leq\lbrack n/2]+1} and for all {\xi\neq0.} Then the multiplier operator {F(D)} on {\mathcal{S}^{\prime}/\mathcal{P}} is bounded from {\mathcal{M}_{p,\mu}^{s}} to {\mathcal{M}_{p,\mu}^{s-m}} and the following estimate hold true

\displaystyle \left\Vert F(D)f\right\Vert _{\mathcal{M}_{p,\mu}^{s-m}}\leq CA\left\Vert f\right\Vert _{\mathcal{M}_{p,\mu}^{s}}, \ \ \ \ \ (14)

where {\mathcal{S}^{\prime}/\mathcal{P}} is the set of equivalence classes in {\mathcal{S}^{\prime}} modulo polynomials with {n} 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 {\varphi} be a Lebesgue integrable function in {\mathbb{R}} and {\alpha\geq 0}, Riemann-Liouville’s integral is defined by

\displaystyle \textbf{I}_{\alpha}\varphi(t)=\int_{0}^{t}R_{\alpha}(t-s)\varphi(s)ds \ \ \ \ \ (15)

and Riemann-Liouville’s derivative by

\displaystyle \textbf{D}^{\alpha}_{t}\varphi= \left(\frac{\partial}{\partial t}\right)^{k}\textbf{I}_{k-\alpha}\varphi(t), \ \ \ \ \ (16)

where {k=\lfloor\alpha\rfloor+1}, {R_{\alpha}(s)=\frac{s^{\alpha-1}}{\Gamma(\alpha)}}, being {\Gamma(\alpha)} denoted by so-called Gamma function.

Hardy and Littlewood give wider properties about this integrals. His showed, for instance, the boundedness of {\textbf{I}_{\alpha}} from {L^{p}(\mathbb{R})} to {L^{q}(\mathbb{R})}, for {1<p<q<\infty}. 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. FujitaSchneider and Wyss, for instance, founds the special function to represent the Green function associated to the linear part of the diffusion-wave problem (FPDE)

\displaystyle \begin{array}{rcl} \mathbf{\partial}_{t}^{\alpha}u =\Delta_{x}u+|u|^{\rho}u,\text{ in }\mathbb{R}^{n},\;t>0 \end{array}

such special function, is knew as Mittag-Leffler function. More precisely,

\displaystyle \mathbb{E}_{\alpha}(z)=\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma(\alpha k +1)} \ \ \ \ \ (17)

and the Green function, defined via Fourier inversion, is given by

\displaystyle \mathcal{K}_{\alpha}(x,t)=\int_{\mathbb{R}^{n}}e^{ix\cdot \xi}\mathbb{E}_{\alpha}(-t^{\alpha}\vert \xi\vert^{2})d\xi, \ \ \ \ \ (18)

which generate the following semigroup of operators {\{G_{\alpha}(t)\}_{t\geq 0}},

\displaystyle G_{\alpha}(t)\varphi=\mathcal{K}_{\alpha}(\cdot,t)\ast\varphi. \ \ \ \ \ (19)

In what follows, we recall some functions which is useful to handle the symbol of the semigroup {G_{\alpha}(t)} (see (19)). For {1<\alpha<2,} let us set

\displaystyle a_{\alpha}(\xi)=|\xi|^{\frac{2}{\alpha}}e^{\frac{i\pi}{\alpha}} ,\;\;\;b_{\alpha}(\xi)=|\xi|^{\frac{2}{\alpha}}e^{-\frac{i\pi}{\alpha}},\text{ for }\xi\in\mathbb{R}^{n}\text{,} \ \ \ \ \ (20)

and

\displaystyle l_{\alpha}(\xi)= \begin{cases} \frac{\sin(\alpha\pi)}{\pi}\int_{0}^{\infty}\frac{|\xi|^{2}s^{\alpha-1}e^{-s} }{s^{2\alpha}+2|\xi|^{2}s^{\alpha}\cos(\alpha\pi)+|\xi|^{4}}ds & \text{ if }\xi\neq0\\ 1-\frac{2}{\alpha}, & \text{ if }\xi=0. \end{cases} \ \ \ \ \ (21)

Lemma 3 Let {1<\alpha<2} and {\mathcal{K}_{\alpha}} be as in (18). We have that

\displaystyle \begin{array}{rcl} \mathbb{E}_{\alpha}(-|\xi|^{2})=\frac{1}{\alpha}(\exp(a_{\alpha}(\xi))+\exp(b_{\alpha}(\xi)))+l_{\alpha}(\xi) \end{array}

and

\displaystyle \frac{\partial^{k}\mathcal{K}_{\alpha}}{\partial x_{i}^{k}}(x,t)=\lambda ^{n+k}\frac{\partial^{k}}{\partial x_{i}^{k}}\mathcal{K}_{\alpha}(\lambda x,\lambda^{\frac{2}{\alpha}}t), \ \ \ \ \ (22)

for all {\;\lambda>0.} Moreover, {\mathcal{K}_{\alpha}(x,t)\geq0}, {P_{\alpha }(|x|,1)=\alpha\mathcal{K}_{\alpha}(x,1)} is a probability measure, and

\displaystyle \begin{array}{rcl} \Vert\mathcal{K}_{\alpha}(\cdot,t)\Vert_{L^{1}(\mathbb{R}^{n})}=\frac {1}{\alpha},\text{ for all }t>0. \end{array}

Proof: Except for (22), all properties contained on the statement can be found in Fujita and Hirata-Miao when {n=1} and {n\geq2,} respectively.

In order to prove (22), we use Fourier inversion and (18) to obtain

\displaystyle \begin{array}{rcl} \frac{\partial^{k}\mathcal{K}_{\alpha}}{\partial x_{i}^{k}}(x,t) &=&\int_{\mathbb{R}^{n}}e^{ix\cdot\xi}(-i\xi_{i})^{k}\mathbb{E}_{\alpha}(-t^{\alpha}|\xi|^{2})d\xi\\ &=&t^{-n\frac{\alpha}{2}}\int_{\mathbb{R}^{n}}e^{ix\cdot\frac{y}{\sqrt{t^{\alpha}}}}(it^{-\frac{\alpha}{2}}y_{i})^{k}\mathbb{E}_{\alpha}(-|y|^{2})dy\\ &=&t^{-\frac{\alpha}{2}(n+k)}\int_{\mathbb{R}^{n}}e^{i\frac{x}{\sqrt{t^{\alpha}}}\cdot y}(iy_{i})^{k}\mathbb{E}_{\alpha}(-|y|^{2})dy\\ &=&t^{-\frac{\alpha}{2}(n+k)}\frac{\partial^{k}\mathcal{K}_{\alpha}}{\partial x_{i}^{k}}(\frac{x}{\sqrt{t^{\alpha}}},1). \end{array}

The desired identity follows by taking {\lambda=1/\sqrt{t^{\alpha}}} in the last equality. \Box

4. Some estimates

The aim of this section is to derive estimates for the semigroup {G_{\alpha}(t).} For that matter we will need pointwise estimates for the fundamental solution {\mathcal{K}_{\alpha}} in Fourier variables.

Lemma 4 Let {1\leq\alpha<2} and {0\leq\delta<2.} There is {C>0} such that

\displaystyle \left\vert \frac{\partial^{k}}{\partial\xi^{k}}\left[ \left\vert \xi\right\vert ^{\delta}E_{\alpha}(-|\xi|^{2})\right] \right\vert \leq C\left\vert \xi\right\vert ^{-\left\vert k\right\vert },\text{ } \ \ \ \ \ (23)

for all {k\in(\mathbb{N}\cup\{0\})^{n}} with {\left\vert k\right\vert \leq\lbrack n/2]+1} and for all {\xi\neq0.}

In the sequel we prove key estimates on Morrey spaces for the semigroup {\{G_{\alpha}(t)\}_{t\geq0}.}

Lemma 5 Let {1\leq\alpha<2}, {1<p\leq q<\infty} , {0\leq\mu<n,} and {\frac{n-\mu}{p}-\frac{n-\mu}{q}<2}. There exists {C>0} such that

\displaystyle \Vert G_{\alpha}(t)f\Vert_{q,\mu}\leq Ct^{-\frac{\alpha}{2}(\frac{n-\mu} {p}-\frac{n-\mu}{q})}\Vert f\Vert_{p,\mu}, \ \ \ \ \ (24)

for all {f\in\mathcal{M}_{p,\mu}.}

Proof: Let {\delta=\frac{n-\mu}{p}-\frac{n-\mu}{q},} {f_{\lambda}(x)=f(\lambda x)} and {h_{\alpha}(x,t)} defined through {\widehat{h_{\alpha}}(\xi,t)=} {\left\vert \xi\right\vert ^{\delta}E_{\alpha }(-|\xi|^{2}t^{\alpha}).} Consider the multiplier operators

\displaystyle \begin{array}{rcl} F(D)f & =&[(-\Delta)^{\frac{\delta}{2}}G_{\alpha}(1)]f=h_{\alpha} (\cdot,1)\ast f(\cdot),\nonumber\\ (\lbrack(-\Delta)^{\frac{\delta}{2}}G_{\alpha}(t)]f)(x) &=&t^{^{-\delta\frac{\alpha}{2}}}\left( h_{\alpha}(\cdot,1)\ast f_{t^{\alpha/2}}(\cdot)\right) _{t^{-\alpha/2}}(x), \end{array}

that is,

\displaystyle F(D)f=t^{^{-\delta\frac{\alpha}{2}}}\left( F(D)(f_{t^{\alpha/2}})\right) _{t^{-\alpha/2}}(x), \ \ \ \ \ (25)

where the symbol of {F(D)} is {\left\vert \xi\right\vert ^{\delta}E_{\alpha }(-|\xi|^{2}t^{\alpha})}. Lemma 4 implies that {F(\xi)} satisfies (13) with {m=0.} Then, we use (5) to obtain

\displaystyle \begin{array}{rcl} \left\Vert \left( F(D)(f_{t^{\alpha/2}})\right) _{t^{-\alpha/2}}\right\Vert _{p,\mu} & =&t^{-\frac{\alpha}{2}(\frac{n-\mu}{p})}\left\Vert F(D)(f_{t^{\alpha/2}})\right\Vert _{p,\mu}\\\ & \leq &Ct^{-\frac{\alpha}{2}(\frac{n-\mu}{p})}\left\Vert f_{t^{\alpha/2} }\right\Vert _{p,\mu}\\ & =&Ct^{-\frac{\alpha}{2}(\frac{n-\mu}{p})}t^{\frac{\alpha}{2}(\frac{n-\mu} {p})}\left\Vert f\right\Vert _{p,\mu} \end{array}

and therefore,

\displaystyle \left\Vert \left( F(D)(f_{t^{\alpha/2}})\right) _{t^{-\alpha/2}}\right\Vert_{p,\mu} \leq C\left\Vert f\right\Vert _{p,\mu}. \ \ \ \ \ (26)

Now, using Sobelev embedding (11), and afterwards (25), we obtain

\displaystyle \begin{array}{rcl} \left\Vert G_{\alpha}(t)f\right\Vert _{q,\mu} & \leq&\left\Vert G_{\alpha }(t)f\right\Vert _{\mathcal{M}_{p,\mu}^{\delta}}\\ & =&\left\Vert (-\Delta)^{\frac{\delta}{2}}G_{\alpha}(t)f\right\Vert _{p,\mu }\\ & =&t^{^{-\delta\frac{\alpha}{2}}}\left\Vert \left( F(D)(f_{t^{\alpha/2} })\right) _{t^{-\alpha/2}}\right\Vert _{p,\mu}\\ & \leq &Ct^{^{-\frac{\alpha}{2}(\frac{n-\mu}{p}-\frac{n-\mu}{q})}}\left\Vert f\right\Vert _{p,\mu}, \end{array}

because of (26). \Box

5. Self-similarity and symmetries for a fractional-wave equation

The problem (1) can be formally converted to the integral equation (see [Hirata-Miao])

\displaystyle u(x,t)={G}_{\alpha}(t)u_{0}(x)+B_{\alpha}(u), \ \ \ \ \ (27)

with

\displaystyle B_{\alpha}(u)(t)=\int_{0}^{t}G_{\alpha}(t-s)\int_{0}^{s}R_{\alpha-1} (s-\tau)|u(\tau)|^{\rho}u(\tau)d\tau ds, \ \ \ \ \ (28)

which should be understood in the Bochner sense in Morrey spaces, being {R_{\eta}(s)=s^{\eta-1}/\Gamma(\eta)}. Throughout this notes a mild solution for (1) is a function {u} 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

\displaystyle \mathbf{\partial}_{t}^{\alpha}u =\Delta_{x}u+|u|^{\rho}u,\text{ in }\mathbb{R}^{n},\;t>0 \ \ \ \ \ (29)

shows that the indexes {k_{1}=2/\rho} and {k_{2}=2/\alpha} are the unique ones such that the function {u_{\lambda}} given by

\displaystyle u_{\lambda}(x,t)=\lambda^{k_{1}}u(\lambda x,\lambda^{k_{2}}t) \ \ \ \ \ (30)

is a solution of that, for each {\lambda>0,} whenever {u} is also. The scaling map for (29) is defined by

\displaystyle u(x,t)\rightarrow u_{\lambda}(x,t). \ \ \ \ \ (31)

Making {t\rightarrow0^{+}} in (31) one obtains the following scaling for the initial condition

\displaystyle u_{0}(x)\rightarrow\lambda^{2/\rho}u_{\lambda}(x). \ \ \ \ \ (32)

Solutions invariant by (31), that is

\displaystyle u(x,t)=u_{\lambda}(x,t)\text{ for all }\lambda>0, \ \ \ \ \ (33)

are called self-similar ones. Since we are interested in such solutions, it is suitable to consider critical spaces for {u(x,t)} and {u_{0}}, i.e., the ones whose norms are invariant by (31) and (32), respectively.

Consider the parameters

\displaystyle \mu=n-2p/\rho\text{ and }\beta=\frac{\alpha}{\rho}-\alpha\frac{n-\mu}{2q}, \ \ \ \ \ (34)

and let {BC((0,\infty),X)} stands for the class of bounded and continuous functions from {(0,\infty)} to a Banach space {X.} We take {u_{0}} belonging to the critical space {\mathcal{M}_{p,\mu}} and study (27) in the Kato-Fujita type space

\displaystyle H_{q}=\{u(x,t)\in BC((0,\infty);\mathcal{M}_{p,\mu})\,:\,t^{\beta}u\in BC((0,\infty);\mathcal{M}_{q,\mu})\}, \ \ \ \ \ (35)

which is Banach with the norm

\displaystyle \Vert u\Vert_{H_{q}}=\sup_{t>0}\Vert u(\cdot,t)\Vert_{p,\mu}+\sup _{t>0}t^{\beta}\Vert u(\cdot,t)\Vert_{q,\mu}. \ \ \ \ \ (36)

Notice that the norm (36) is invariant by scaling transformation (31).

From Lemma 1, a typical data belonging to {\mathcal{M}_{p,\mu}} is the homogeneous function

\displaystyle u_{0}(x)=\,{\Omega(}\frac{x}{\left\vert x\right\vert }{)}/\left\vert x\right\vert {^{-\frac{2}{\rho}}} \ \ \ \ \ (37)

where {1\leq p<\frac{n\rho}{2}} and {\Omega} is a bounded function on sphere {\mathbb{S}^{n-1}.} 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 {0<\rho<\infty}, {1<\alpha<2}, {1<p<\frac {n\rho}{2}}, and {\mu=n-{2p}/{\rho}}. Suppose that {\frac{n-\mu}{p} -\frac{n-\mu}{q}<2} and

\displaystyle 1-\frac{1}{\alpha}\,\frac{\rho}{\rho+1}<\frac{p}{q}<\frac{1}{\alpha}\;\;\text{ and }\;\frac{\rho p}{p-1}<q<\infty. \ \ \ \ \ (38)

  • (i) (Existence and uniqueness) There exist {\ \varepsilon>0} and {\delta=\delta(\varepsilon)} such that if {\Vert u_{0}\Vert_{p,\mu}\leq\delta} then the equation (1) has a mild solution {u\in H_{q}}, which is the unique one in the ball {D_{2\varepsilon}=\{u\in H_{q};\left\Vert u\right\Vert _{H_{q}}\leq2\varepsilon\}}. Moreover, {u\rightharpoonup u_{0}} in {\mathcal{D}^{\prime}(\mathbb{R}^{n})} as {t\rightarrow0^{+}}
  • (ii) (Continuous dependence on data) Let {\mathcal{I}_{0}=\{u_{0} \in\mathcal{M}_{p,\mu};\left\Vert u\right\Vert _{p,\mu}\leq\delta\}.\,\ }The data-solution map is Lipschitz continuous from {\mathcal{I}_{0}} to {D_{2\varepsilon}}.

Remark 1

  • (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 {f(u)} instead of {u\left\vert u\right\vert ^{\rho}}, where {f\in C(\mathbb{R})}, {f(0)=0} and there is {C>0} such that

    \displaystyle |f(a)-f(b)|\leq C|a-b|(|a|^{\rho}+|b|^{\rho}),\text{ for all }a,b\in \mathbb{R}.

  • (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 {T>0}. Here we should consider the local-in-time space\displaystyle H_{q,T}=\{u(x,t)\in BC((0,T);\mathcal{M}_{p,\mu})\,:\,\lim\sup_{t\rightarrow 0^{+}}t^{\beta}\Vert u(\cdot,t)\Vert_{q,\mu}=0\}, and {u_{0}\in\mathcal{M}_{p,\mu}} such that {\lim\sup_{t\rightarrow0^{+} }t^{\beta}\Vert G_{\alpha}(t)u_{0}\Vert_{q,\mu}=0.} In particular, this condition is verified when {u_{0}} belongs to the closed subspace {\ddot{\mathcal{M}}_{p,\mu}} (see (8)).

Let {O(n)} be the orthogonal matrix group in {\mathbb{R}^{n}} and let {\mathcal{G}} be a subset of {O(n).} A function {h} is said symmetric and antisymmetric under the action of {\mathcal{G}} when {h(x)=h(Mx)} and {h(x)=-h(M^{-1}x)}, respectively, for every {M\in\mathcal{G}}.

Theorem 7 Under the hypotheses of Theorem 6.

  • (i) (Self-similarity) If {u_{0}} is a homogeneous function of degree {-\frac{2}{\rho}}, then the mild solution given in Theorem 6 is self-similar.
  • (ii) (Symmetry and antisymmetry) The solution {u(x,t)} is antisymmetric (resp. symmetric) for {t>0}, when {u_{0}} is antisymmetric (resp. symmetric) under {\mathcal{G}.}
  • (iii) (Positivity) If {u_{0}\not \equiv 0} and {u_{0}(x)\geq0} (resp. {u_{0}(x)\leq0}) then {u} is positive (resp. negative).

Remark 2 (Special examples of symmetry and antisymmetry)

  • (i) The case {\mathcal{G}=O(n)} corresponds to radial symmetry. Therefore, it follows from Theorem 7 (ii) that if {u_{0}} is radially symmetric then {u(x,t)} is radially symmetric for {t>0}.
  • (ii) Let {Mx=-x} be the reflection over the origin and let {I_{\mathbb{R}^{n}}} be the identity map. The case {\mathcal{G} =\{I_{\mathbb{R}^{n}},M\}} corresponds to parity of functions, that is, {h(x)} is even and odd when {h(x)=h(-x)} and {h(x)=-h(-x)}, respectively. So, from Theorem 7 (ii), we have that the solution {u(x,t)} is even (resp. odd) for {t>0,} when {u_{0}(x)} is even (resp. odd).

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