On the uniqueness of mild solutions for the parabolic-elliptic Keller-Segel system in the critical $L^{p}$-space

1.   Introduction

We are concerned with the parabolic-elliptic Keller-Segel (or Patlak-Keller-Segel) system

where $0 < T\leq\infty$, $u(x, t)\geq0$ represents the density of cells and $v(x, t)\geq0$ is the concentration of the chemoattractant. The parameters $\gamma\geq0$ and $\kappa > 0$ denote the decay and production rate of the chemoattractant, respectively. The model works for $n\geq2$ but, as will be seen later, we restrict themselves to $n\geq4$ due to technical issues.

System (1.1) is a chemotaxis model with a mathematical structure of parabolic-elliptic type. It is used in the study of aggregation of biological species, describing the behavior of organisms (e.g., bacteria) that move towards high concentration of a chemical secreted by themselves.

In view of (1.1)$_{2},$ we can express $v = \kappa(-\Delta+\gamma I)^{-1}u$ and, according to Duhamel's principle, the Cauchy problem (1.1) can be formally converted to the integral equation

where $G(t) = e^{\Delta t}$ stands for the heat semigroup and the bilinear term $B$ is given by

Functions $u(x, t)$ satisfying (1.2) are called mild solutions for (1.1). Here the mild formulation (1.2) is considered in a suitable dual sense, see Section 3 for details.

For $\gamma = 0,$ Eqs. (1.1)$_{1}$–(1.1)$_{2}$ has the scaling

which, for the initial data, induces

Spaces invariant under the scaling (1.5), namely $\left\Vert u_{0}\right\Vert _{X}\approx\left\Vert \lambda^{2}u_{0}(\lambda x)\right\Vert _{X}$ for all $\lambda > 0$, are called critical spaces for (1.1).

In the dimension $n = 2,$ it is well-known that there exists a threshold value for the initial mass $M = \int u_{0}dx$ that decides if solutions exist globally ($M < 8\pi/\kappa$) or blow up in a finite time ($M > 8\pi/\kappa$) (see, e.g., ^[6,7]). Note that the space $L^{1} (\mathbb{R}^{2})$ is critical for (1.1). For $n\geq3,$ one might wonder if some critical space could play a similar role as the $L^{1}$-space in $n = 2$ (for example, $L^{\frac{n}{2}}(\mathbb{R}^{n})$), however, it is still an open problem to know whether there exists such a suitable space. In connection with that, in dimensions $n\geq3,$ there is a huge literature about existence of mild solutions for (1.1) and its parabolic-parabolic version with smallness conditions on the initial data in critical spaces. Without making a complete list, we mention the results in $L^{1}\cap L^{\frac{n}{2}\text{ }}$^[10], $L^{\frac{n}{2}}$ ^[22] (weak solutions), Marcinkiewicz $L^{\frac{n}{2}, \infty}$ (weak$-L^{p}$ spaces) ^[23], $\mathcal{PM}^{n-2}$ ^[3], Besov $\dot {B}_{q, \infty}^{\frac{n}{q}-2}$ ^[18], Triebel-Lizorkin $\dot {F}_{\infty, 2}^{-2}$ ^[19], Morrey $\mathcal{M}_{q, n-2q}$ ^[2], Fourier-Besov $F\dot{B}_{1, r}^{-2}$ ^[18], Besov-Morrey $\mathcal{N}_{q, \mu, \infty}^{\frac{n-\mu}{q}-2}$ ^[13], and Fourier-Besov-Morrey spaces $\mathcal{FN}_{q, \mu, \infty}^{n-2-\frac{n-\mu }{q}}$^[9], among others. It is worth noting that most of the above existence results of small mild solutions in critical spaces were inspired by those for Navier-Stokes equations, see, e.g., ^{[1,11,15,20,21,24,25,30]}, and their references.

On the other hand, the uniqueness in critical spaces $X$ is more subtle and needs some care. For $n\geq3,$ most of the above existence results are proved by constructing a fixed point argument in time-dependent spaces with norms composed of two ou more parts. One is the norm of the persistence space $L^{\infty}((0, \infty); X)$ and the others are auxiliary norms such as Kato time-weighted type norms, time-spatial mixed Lebesgue norms (i.e., $L^{q}((0, T);L^{p})$-norms or, more generally, $L^{q}((0, T);Y)$ with $q\neq\infty$) and Chemin-Lerner type norms, which are used to control the bilinear term $B(u, w).$ Also, solutions are continuous at $t > 0$ but only time-weakly continuous at $t = 0^{+}$, since the heat semigroup $\{e^{t\Delta }\}_{t\geq0}$ is not strongly continuous at $t = 0^{+}$. This lack of continuity can be overcome by considering either the maximal subspace $\tilde{X}$ in which $\{e^{t\Delta}\}_{t\geq0}$ is continuous or the closure of $C_{0}^{\infty}(\mathbb{R}^{n})$ in $X$, and then solutions belong to $C([0, T);\tilde{X})$ with large initial data $u_{0}\in\tilde{X}$ and small $T > 0.$ The estimates involving auxiliary norms in the proof of existence results, in principle, provide only a conditional uniqueness result, that is, uniqueness in a space more restricted than the natural one $C([0, T);\tilde {X}).$ For the sake of completeness, in the case $n = 2$ we would like to mention the uniqueness results of weak/mild solutions for (1.1) (and its parabolic-parabolic version) with finite mass, finite second moment and finite entropy (see, e.g., ^[8,12,27] and their references).

A way to obtain unconditional uniqueness in the critical class $C([0, T);\tilde {X})$ (or uniqueness of small solutions in $L^{\infty}((0, \infty); X)$) is to prove the bilinear estimate

where $C > 0$ is a constant. This approach has already been employed in the context of Navier-Stokes equations. For example, see ^{[11,14,25,26,28,30]} to results in the framework of critical Lebesgue, Marcinkiewicz, Morrey and weak-Morrey spaces.

Next, let us discuss in more detail the works ^[22] and ^[23], which are more directly related to our results. In ^[23], Kozono-Sugiyama proved local well-posedness of mild solutions for (1.1) with small data $u_{0}\in L^{\frac{n}{2}, \infty}$ and $n\geq3,$ where the existence and uniqueness are obtained in the class

where $u$ is time-weakly continuous at $t = 0^{+}$ and $\beta = 1-\frac{n}{2q}.$ Also, $u\in BC([0, T);L^{\frac{n}{2}})$ provided that $u_{0}\in L^{\frac{n}{2} }\hookrightarrow L^{\frac{n}{2}, \infty}$. They employed a point-fixed argument by using Kato's approach (see ^[20]) and their bilinear estimates are performed with the help of the auxiliary time-weighted norm $\sup_{t\in (0, T)}t^{\alpha}\left\Vert u(\cdot, t)\right\Vert _{L^{q}}.$ The solution $u$ is global ($T = \infty$) if $\left\Vert u_{0}\right\Vert _{L^{\frac{n}{2}, \infty}}$ is small enough. For the uniqueness, besides assuming (1.7), it is required a smallness condition in the auxiliary norm, that is, the uniqueness is obtained in a space more restricted than $BC((0, T);L^{\frac{n}{2}, \infty}).$ Moreover, assuming additional regularity on the initial data, they obtained the existence of local (or small global) strong $L^{p}$-solutions.

In ^[22], Kozono-Sugiyama-Yahagi proved the existence of local weak solutions $u$ with respect to a suitable set of test functions. More precisely, for initial data $u_{0}\in L^{\frac{n}{2}}(\mathbb{R}^{n}),$ they obtained local-in-time solutions in

where $T > 0$ is small enough, $n\geq3,$ $2/q+n/p = 2$, $2 < q < \infty,$ $n/2 < p < n,$ and $p\leq n^{2}/2(n-2).$ These weak solutions can be extended globally ($T = \infty$) if $\left\Vert u_{0}\right\Vert _{L^{\frac{n}{2}}}$ is small enough. They constructed solutions via an approximation scheme of strong solutions whose existence was obtained in ^[23]. The uniqueness was obtained in the class $L^{q}((0, T);L^{p}(\mathbb{R}^{n}))$ with $n\geq3,$ $2/q+n/p = 2$ and $n/2 < p < n.$ For $n\geq4,$ the uniqueness in the natural persistence space $C([0, T);L^{\frac{n}{2}}(\mathbb{R}^{n}))$ was obtained without any further condition, while the uniqueness in $L^{\infty }((0, T);L^{\frac{n}{2}}(\mathbb{R}^{n}))$ required small conditions on $u$ and $u_{0}.$ For that, Kozono-Sugiyama-Yahagi converted the uniqueness problem to the one of obtaining global strong solutions for the associated adjoint equation, where coefficients depend on weak solutions, by using maximal regularity of the heat equation and suitable estimates involving $L^{q}((0, T);L^{p}(\mathbb{R}^{n}))$-norms with $q\neq\infty.$

As pointed out by the authors of ^[22], it is not clear whether mild solutions satisfy their weak formulation. So, a natural question is to know if we have unconditional uniqueness of mild solutions for (1.1) in $C([0, T);L^{\frac{n}{2}}(\mathbb{R}^{n})),$ as well as the uniqueness in $L^{\infty}((0, T);L^{\frac{n}{2}}(\mathbb{R}^{n}))$ with $\sup_{t\in(0, T)}\left\Vert u(\cdot, t)\right\Vert _{L^{\frac{n}{2}}}$ small enough. Another question is to know if it is possible to obtain these uniqueness properties (and also existence and continuous dependence on initial data) without using auxiliary norms such as, for example, those in ^[22] and ^[23].

In this paper we obtain positive answers for the above questions for $n\geq4$. First, inspired by ^[30], we prove estimate (1.6) with $X = L^{\frac{n}{2}, \infty}(\mathbb{R}^{n})$ by means of careful estimates on the predual space of $L^{\frac{n}{2}, \infty}(\mathbb{R}^{n})$, that is, the Lorentz space $L^{\frac{n}{n-2}, 1}(\mathbb{R}^{n})$ (see Section 2 for the definition). So, adapting arguments found in ^[28], we obtain the unconditional uniqueness of mild solutions in the class $C([0, T);L^{\frac {n}{2}}(\mathbb{R}^{n}))$ with $0 < T\leq\infty$ and initial data $u_{0}\in L^{\frac{n}{2}}(\mathbb{R}^{n})$ (see Theorem 3.1), because $L^{\frac{n}{2}}(\mathbb{R}^{n})$ is contained in $\tilde{X}$ (maximal subspace of $L^{\frac{n}{2}, \infty}(\mathbb{R}^{n})$ where $\{e^{t\Delta }\}_{t\geq0}$ is continuous). Moreover, using heat semigroup estimates and (1.6), we have the well-posedness of small solutions in $L^{\infty }((0, T);L^{\frac{n}{2}, \infty}(\mathbb{R}^{n}))$ with $0 < T\leq\infty$. Since we have the continuous inclusion $L^{\frac{n}{2}}\hookrightarrow$ $L^{\frac {n}{2}, \infty}$, the uniqueness of mild solutions in $L^{\infty} ((0, T);L^{\frac{n}{2}}(\mathbb{R}^{n}))$ holds true provided that $\sup _{t\in(0, T)}\left\Vert u(\cdot, t)\right\Vert _{L^{\frac{n}{2}}}$ is small enough. Furthermore, we analyze the dependence of the well-posedness result with respect to the decay rate $\gamma$ of the chemoattractant (see Remark 3.2).

Finally, we observe that our results work well by considering non-negative $u_{0}$ and $u$ in (1.1) as well as without any sign restrictions on them. However, we have considered the former for physical reasons associated to the model.

This paper is organized as follows. In Section 2, we give some preliminaries about Lorentz spaces and, in particular, weak-$L^{p}$ spaces. Section 3 is dedicated to the statements of our results and some further remarks. The proofs of results are performed in Section 4.

2.   Preliminaries

This section is devoted to recalling some basic definitions and properties about Lorentz spaces.

Let $|\cdot|$ stands for the Lebesgue measure and let $\lambda_{f} (s) = \left\vert \{x\in\mathbb{R}^{n}:|f(x)| > s\}\right\vert$ be the distribution function of a measurable function $f:\mathbb{R}^{n} \rightarrow \mathbb{R}.$ The rearrangement of $f$ is defined as

The Lorentz space is the complete quasi-normed space

where the quasi-norm $\left\Vert f\right\Vert _{p, d}^{\ast}$ is given by

Taking $d = p$ and $d = \infty$ in (2.2), we obtain the Lebesgue space $L^{p, p}(\mathbb{R}^{n}) = L^{p}(\mathbb{R}^{n})$ and the Marcinkiewicz space $L^{p, \infty},$ also called weak-$L^{p}$, respectively. The case $p = \infty$ and $1\leq d < \infty$ was removed from (2.3) because $L^{\infty, d}$ is the trivial space.

In general, the quantity (2.3) is not a norm on $L^{p, d}$. Consider the double-rearrangement

and define the quantity $\Vert\cdot\Vert_{p, d}$ exactly as in (2.3) but replacing $f^{\ast}$ with $f^{\ast\ast}.$ For $1 < p\leq\infty,$ the quantities $\Vert\cdot\Vert_{p, d}$ and $\Vert\cdot \Vert_{p, d}^{\ast}$ are topologically equivalent on $L^{p, d}$, since we have

The pair $(L^{p, r}, \Vert\cdot\Vert_{p, d})$ is a Banach space and, unless mentioned otherwise, we consider it when $1 < p\leq\infty.$ For $0 < p\leq1$, $L^{p, d}$ is endowed with $\Vert\cdot\Vert_{p, d}^{\ast}$. In the case $p = d = 1$ we have $\Vert\cdot\Vert_{1, 1}^{\ast}$ coincides with the standard $L^{1}$-norm and $L^{1, 1} = L^{1}$.

Lorentz spaces have the scaling property

For $1\leq d_{1}\leq p\leq d_{2}\leq\infty$ and $1\leq p\leq\infty$, we have the continuous strict inclusions

and then $L^{p, d}$ becomes larger as the second index $d$ goes from $1$ to $\infty$.

Next we recall the Hölder inequality in Lorentz spaces (see ^[17,29]). Let $1 < p_{1}, p_{2}, p_{3}\leq\infty$ and $1\leq d_{1}, d_{2}, d_{3}\leq\infty$ be such that $1/p_{3} = 1/p_{1}+1/p_{2}$ and $1/d_{1} +1/d_{2}\geq1/d_{3}.$ Then, there exists a constant $C > 0$ (independent of $f$ and $g$) such that

For $1\leq p, d < \infty$ (with $d = 1$ when $p = 1$), we have that the dual space of $L^{p, d}$ is $L^{p^{\prime}, d^{\prime}}$(see ^[16]). In particular, the dual of $L^{p, 1}$ is $L^{p^{\prime}, \infty}$ for $1\leq p < \infty.$ The space of compactly-supported smooth functions $C_{0}^{\infty}(\mathbb{R}^{n})$ is dense in $L^{p, d}(\mathbb{R}^{n})$ for $1\leq p, d < \infty.$

Young inequality works well in the framework of Lorentz spaces. In fact, if $1 < p_{1}, p_{2}, p_{3}\leq\infty$ and $1\leq d_{1}, d_{2}, d_{3}\leq\infty$ with $1/p_{3} = 1/p_{1}+1/p_{2}-1$ and $1/d_{1}+1/d_{2}\geq1/d_{3},$ then (see ^[29])

where $C > 0$ is a universal constant. Moreover, for $p_{1} = 1$ and $1 < p = p_{2} = p_{3}\leq\infty$, we have the inequality (see ^[4])

Lorentz spaces has nice interpolation properties. For that, recall that the functor $(\cdot, \cdot)_{\theta, r}$ constructed via the $K_{\theta, q}$-method is exact of exponent $\theta$ on the categories of quasi-normed and normed spaces. Let $0 < p_{1} < p_{2}\leq\infty, \ 0 < \theta < 1, \ \frac{1}{p} = \frac {1-\theta}{p_{1}}+\frac{\theta}{p_{2}}$ and $1\leq d_{1}, d_{2}, d\leq\infty$. Employing $(\cdot, \cdot)_{\theta, r}$ in Lorentz spaces, we obtain (see [5,Chapter 5])

For $1 < p\leq\infty$ and $1\leq d\leq\infty$, by interpolating (2.8), we arrive at the inequality

where $C > 0$ is a universal constant.

3.   Main results

In this section we state the bilinear estimate (1.6) in weak- $L^{\frac{n}{2}}$ and uniqueness result. Also, we make some comments about global existence and its dependence on the decay rate $\gamma$ of the chemoattractant, non-negativity and mass conservation.

Before proceeding, we point out that the mild formulation (1.2) and its bilinear term should be meant in a suitable dual sense in the $L^{\frac{n} {2}, \infty}$-setting by using its predual space $L^{\frac{n}{n-2}, 1}$ and the duality pairing $\langle f, g\rangle = \int_{\mathbb{R}^{n}}fgdx$ (see ^[30]). More precisely, for $u, w\in L^{\infty}((0, T);L^{\frac {n}{2}, \infty})$ we define $B(u, w)$ by

for all $\phi\in L^{\frac{n}{n-2}, 1}(\mathbb{R}^{n})$ and a.e. $t\in(0, T).$ Note also that, for $u_{0}\in L^{\frac{n}{2}, \infty}(\mathbb{R}^{n}),$ the convolution $G(t)u_{0}$ is well-defined and

Thus, the formulation (1.2) should be meant as

for all $\phi\in L^{\frac{n}{n-2}, 1}(\mathbb{R}^{n})$ and a.e. $t\in(0, T).$ In other words, taking into account (3.2), $u$ verifies (1.2) with $B(u, u)$ given by (3.1).

Our results read as follows.

Theorem 3.1. Let $n\geq4, \gamma\geq0,$ $\kappa > 0$ and $0 < T\leq\infty.$

(i) (Bilinear estimate) Let $B(\cdot, \cdot)$ be the bilinear form (1.3). There exists a constant $K > 0$ (independent of $\gamma$) such that

for all $u, v\in L^{\infty}((0, T);L^{\frac{n}{2}, \infty}(\mathbb{R}^{n}))$, where $g(\gamma) = 1$ and $g(\gamma) = \gamma^{-(n-1)}$ if $\gamma = 0$ and $\gamma > 0,$ respectively.

(ii) (Uniqueness) Let $u$ and $w$ be mild solutions of (1.1) in the class $C([0, T);L^{\frac{n}{2}}(\mathbb{R}^{n}))$ with the same initial data $u_{0}$. Then $u(\cdot, t) = w(\cdot, t)$ in $L^{\frac{n}{2}}(\mathbb{R} ^{n}),$ for all $t\in\lbrack0, T)$.

Remark 3.2. (Further comments)

(i) (The restriction $n\geq4$) Due to the duality arguments in our proofs, we need to employ Lemma 4.3 with $p = \frac{n}{2}$ and $r = \frac{n}{3} > 1,$ and then $n\geq4.$ In turn, we need $r > 1$ for Hölder's inequality (2.6) and the quantity $\left\Vert \cdot\right\Vert _{r, \infty}$ in $L^{r, \infty}$ to be a norm. An interesting question would be to know whether, with some suitable relaxed integrability conditions, those arguments could be adapted to obtain a similar uniqueness result in lower dimensions.

(ii) (Well-posedness) Let $0 < T\leq\infty,$ $\gamma\geq0,$ $n\geq4$ and $u_{0}\in$ $L^{\frac{n}{2}, \infty}(\mathbb{R}^{n})$. Under a small assumption on $\left\Vert u_{0}\right\Vert _{\frac{n}{2}, \infty}$ and a fixed-point argument, Theorem 3.1 together with heat semigroup estimates (see (4.1)) imply the well-posedness of small mild solutions in the class $L^{\infty}((0, T);L^{\frac{n}{2}, \infty}(\mathbb{R}^{n})).$ In particular, we obtain the uniqueness of sufficiently small mild solutions in $L^{\infty }((0, T);L^{\frac{n}{2}, \infty}(\mathbb{R}^{n}))$ or, using the continuous inclusion $L^{\frac{n}{2}}(\mathbb{R}^{n})\hookrightarrow L^{\frac{n} {2}, \infty}(\mathbb{R}^{n}),$ in $L^{\infty}((0, T);L^{\frac{n}{2}} (\mathbb{R}^{n})).$ Moreover, the solution $u$ belongs to $BC((0, T);L^{\frac {n}{2}, \infty}(\mathbb{R}^{n}))$ with time-weak continuity at $t = 0^{+}.$ For $u_{0}\in L^{\frac{n}{2}}(\mathbb{R}^{n})$, one obtains $u\in BC([0, T);L^{\frac{n}{2}}(\mathbb{R}^{n}))$ and the smallness condition on $u_{0}$ can be replaced with a smallness one on the existence-time $T > 0$, and $u\in$ $C([0, T);L^{\frac{n}{2}}(\mathbb{R}^{n}))$ can be large. For $\gamma = 0$ and $T = \infty,$ the obtained solution is self-similar provided that $u_{0}$ is homogeneous of degree $-2$.

(iii) (Non-negativity and mass conservation) Due to the fixed-point argument, the solution $u$ in item (ii) of this remark can be obtained as the limit of the Picard sequence $u^{(k+1)} = u^{(1)}+B(u^{(k)}, u^{(k)}),$ $k\in\mathbb{N}$, and $u^{(1)} = G(t)u_{0}.$ Let $u_{0}\in L^{\frac{n}{2}, \infty}(\mathbb{R}^{n})$ be non-negative. Using the parabolic regularization of the heat semigroup (see, e.g., (4.1)), an induction procedure, and the divergence structure of $B(\cdot, \cdot)$, one can show that $u^{(k)}$ is smooth and non-negative, for each $k$. Since the convergence $u^{(k)} \rightarrow u$ in $L^{\infty}((0, T);L^{\frac{n}{2}, \infty}(\mathbb{R}^{n}))$ preserves non-negativity, it follows that $u$ is non-negative. Furthermore, for $u_{0}\in L_{+}^{1}(\mathbb{R}^{n})\cap L^{\frac{n}{2}, \infty}(\mathbb{R}^{n}),$ using (4.1) and reducing the size of $\left\Vert u_{0}\right\Vert _{\frac {n}{2}, \infty}$ (if necessary), one can show further integrability properties and polynomial time decay of $L^{p}$-norms for $u(\cdot, t)$ and $B(\cdot, t),$ for $t > 0,$ and then obtain $u(\cdot, t)\in L_{+}^{1}$ and $B(u, u)(\cdot, t)\in L^{1}.$ After, using the divergence form of $B(\cdot, \cdot),$ one can obtain the mass conservation of the solution, that is, $\int_{\mathbb{R}^{n}} u(\cdot, t)dx = \int_{\mathbb{R}^{n}}u_{0}dx$, for $t > 0$.

(iv) (Large decay rate of the chemoattractant) Let $0 < T\leq\infty.$ Considering $\gamma > 0$ large enough, we can make $\kappa g(\gamma)K$ small enough and then obtain the well-posedness of mild solutions for (1.1) in $L^{\infty}((0, T);L^{\frac{n}{2}, \infty}(\mathbb{R}^{n})),$ without smallness conditions on the existence-time $T$ and initial data $u_{0}.$ More precisely, in order to employ a fixed-point argument, we need $4\left\Vert u_{0} \right\Vert _{\frac{n}{2}, \infty}\left(\kappa g(\gamma)K\right) < 1$ which leads us to

4.   Proof of Theorem 3.1

4.1.   Proof of item (i) (Bilinear estimate)

We start with a lemma that will be useful to handle the coupling operator in (1.1) in Lorentz spaces.

Lemma 4.1. Let $n\geq2,$ $1 < p < n$, $1\leq d\leq\infty$ and $\frac{1} {q} = \frac{1}{p}-\frac{1}{n}.$ The operator $L_{j} = \partial_{j}(-\Delta+\gamma I)^{-1}$ is continuous from $L^{p, d}(\mathbb{R}^{n})$ to $L^{q, d} (\mathbb{R}^{n})$, for each $j = 1, 2, ..., n.$ Moreover, for $\gamma > 0$, there exists a constant $C > 0$ (independent of $f$ and $\gamma$) such that

Proof. We can write the multiplier operator $L_{j}$ as

Taking $\gamma = 0$ and $\gamma = 1,$ we have that

For $\gamma = 0,$ by using Young's inequality in Lorentz space (2.7) with $p_{3} = q,$ $p_{1} = \frac{n}{n-1}$, and $p_{2} = p,$ we obtain that

Next we deal with the case $\gamma > 0.$ By a scaling argument, note that $\widehat{\mathcal{K}}_{j, \gamma}(\xi) = \gamma^{-1/2}\widehat{\mathcal{K} }_{j, 1}(\gamma^{-1/2}\xi)$ and then

Thus, again using (2.7) with the same indexes above, it follows that

In the lemma below, we recall some known estimates in Lorentz spaces for the heat semigroup (see ^[30]).

Lemma 4.2. (i) Let$\ m\in\{0\}\cup\mathbb{N},$ $1 < r\leq p\leq\infty$, and $1\leq d_{1}, d_{2}\leq\infty.$ Then, there exists a constant $C > 0$ such that

(ii) (Yamazaki's estimate) Let $1 < r < p < \infty$ $.$ There is a constant $C > 0$ such that

In the next lemma, by means of a duality argument and (4.2), we provide estimates for the linear operator

which is linked to "Duhamel structure" of (1.3). Just like (1.3), the operator (4.3) is understood in the sense of duality, as explained in Section 3. Note that the lemma is valid for $n\geq2.$

Lemma 4.3. Let $n\geq2$ and $1 < r < p < \infty$ be such that $\frac{n} {r}-\frac{n}{p} = 1$. There exists a constant $C > 0$ such that

for all $f\in L^{\infty}((0, \infty),$ $L^{r, \infty})$, where the supremum over $t > 0$ is taken in the essential sense.

Proof. First, denoting the heat kernel by $\varsigma(x, t),$ we have that $G(t)\phi =$ $\varsigma(\cdot, t)\ast\phi(\cdot)$. A duality argument and Hölder's inequality (2.6) allow us to estimate the $L^{p, \infty}$-norm of $\mathcal{Q}(f)$ as

Now note that $1 < p^{\prime} < r^{\prime} < \infty$ and that the condition $\frac{n}{r}-\frac{n}{p} = 1$ implies

Using (4.6) and (4.2) in order to estimate the integral in the right-hand side of (4.5), we arrive at

as required.

Proof of item (i). Let $0 < T\leq\infty.$ For each $t\in(0, T),$ consider $f_{t}(x, s)$ given by

and note that (1.3) can be rewritten as

Taking $p = \frac{n}{2}$, $l = n$ and $r = \frac{n}{3},$ observe that $\frac{1} {r} = \frac{1}{p}+\frac{1}{l}$ and $\frac{1}{l} = \frac{1}{p}-\frac{1}{n}.$ Then, we can employ Hölder's inequality and Lemma 4.1 to estimate

where $g(\gamma) = 1$ and $g(\gamma) = \gamma^{-(n-1)}$ if $\gamma = 0$ and $\gamma > 0,$ respectively. It follows that $f_{t}\in L^{\infty}((0, \infty), L^{r, \infty}(\mathbb{R}^{n})),$ for all $t\in(0, T).$

Now, noting that

using Lemma 4.3 and afterwards (4.7), we arrive at

4.2.   Proof of item (ii) (Uniqueness)

With the bilinear estimate (3.3) in hands, the uniqueness of solutions in $C([0, T);L^{\frac{n}{2}, \infty})$ follows by adapting an argument due to Meyer ^[28] for our mild formulation (see also ^[11,25]).

Let $u$ and $w$ be mild solutions of (1.1) in $C([0, T);L^{\frac{n}{2} }(\mathbb{R}^{n}))$ such that $u(0) = w(0) = u_{0}\in L^{\frac{n}{2}} (\mathbb{R}^{n})$. We claim that there exists $0 < T_{1} < T$ such that $u(\cdot, t) = w(\cdot, t)$ in $L^{\frac{n}{2}}(\mathbb{R}^{n}),$ for all $t\in\lbrack0, T_{1})$. Considering $h = u-w,$ $h_{1} = G(t)u_{0}-u$ and $h_{2} = G(t)u_{0}-w,$ we can rewrite the difference of the quadratic terms inside (1.3) as follows

Thus, we can estimate $h$ as

Using (3.3), the parcel $J_{1}(t)$ can be estimated as

Next we turn to $I_{2}$. Take $1 < l < \frac{n}{2} < \beta < n$ and $b > \beta$ satisfying $\frac{1}{l} = \frac{2}{n}+\frac{1}{b},$ $\frac{1}{b} = \frac{1}{\beta }-\frac{1}{n}$ and let $\eta_{\beta} = 1-\frac{n}{2\beta}.$ Also, note that $\frac{1}{l} = \frac{1}{\beta}+\frac{1}{n}$ and $\frac{1}{n} = \frac{2}{n} -\frac{1}{n}.$ Then, using (4.1), Hölder's inequality and afterwards (4.1), we arrive at

for all $t\in(0, T)$, where we used above that $-\frac{n}{2\beta}-\eta_{\beta }+1 = 0$ and

Inserting (4.9)–(4.10) into (4.8) yields

where

Using that $G(t)u_{0}\rightarrow u_{0}$ and $u, v\rightarrow u_{0}$ in $L^{\frac{n}{2}}(\mathbb{R}^{n}),$ as $t\rightarrow0^{+},$ and $L^{\frac{n} {2}}(\mathbb{R}^{n})\hookrightarrow L^{\frac{n}{2}, \infty}(\mathbb{R}^{n}),$ we obtain

Moreover, since $u_{0}\in L^{\frac{n}{2}}(\mathbb{R}^{n}),$ there exists a sequence $\{u_{0, k}\}_{k}\subset L^{\frac{n}{2}}(\mathbb{R}^{n})\cap L^{\beta, \infty}(\mathbb{R}^{n})$ such that $u_{0, k}\rightarrow u_{0}$ in $L^{\frac{n}{2}}(\mathbb{R}^{n})\hookrightarrow L^{\frac{n}{2}, \infty }(\mathbb{R}^{n}).$ In fact, it is sufficient to take $u_{0, k} = G(\frac{1} {k})u_{0}$ and use (4.1). Then, we can estimate

In view of (4.12), (4.13) and (4.14), we can take $T = T_{1} > 0$ small enough such that $CA(T_{1}) < 1.$ Now, estimate (4.11) implies that $h(\cdot, t) = 0$ for all $t\in$ $[0, T_{1}),$which gives the desired claim.

Finally, we are going to show that the smallness condition on $T_{1}$ can be removed. For that, consider

If $T^{\ast} = T,$ then $u = w$ in $[0, T),$ as desired. If $T^{\ast} < T\leq\infty$, we have that $u(\cdot, t) = w(\cdot, t)$ for all $t\in\lbrack0, T^{\ast})$. By time-continuity, we obtain that $u(\cdot, T^{\ast}) = w(\cdot, T^{\ast})$ and then, by the first part, there exists a sufficiently small $\rho > 0$ such that $u(\cdot, t) = w(\cdot, t)$ for all $t\in\lbrack T^{\ast}, T^{\ast}+\rho)$. So, $u(\cdot, t) = w(\cdot, t)$ in $[0, T^{\ast}+\rho)$, which contradics the maximality of $T^{\ast} < \infty$.

Acknowledgments

L. C. F. Ferreira was supported by FAPESP and CNPQ, Brazil.

Conflict of interest

The author declares no conflict of interest.