A doubly critical semilinear heat equation in the $L^1$ space

Abstract

We study the existence and nonexistence for a Cauchy problem of the semilinear heat equation:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tu=\Delta u+|u|^{p-1}u &{} \text {in}\ \mathbb {R}^N\times (0,T),\\ u(x,0)=\phi (x) &{} \text {in}\ \mathbb {R}^N\end{array}\right. } \end{aligned}$$

in $L^1(\mathbb {R}^N)$. Here, $N\ge 1$, $p=1+2/N$ and $\phi \in L^1(\mathbb {R}^N)$ is a possibly sign-changing initial function. Since $N(p-1)/2=1$, the $L^1$ space is scale critical and this problem is known as a doubly critical case. It is known that a solution does not necessarily exist for every $\phi \in L^1(\mathbb {R}^N)$. Let $X_q:=\{\phi \in L^1_\mathrm{{loc}}(\mathbb {R}^N)\ |\ \int _{\mathbb {R}^N}|\phi |\left[ \log (e+|\phi |)\right] ^q\mathrm{d}x<\infty \}(\subset L^1(\mathbb {R}^N))$. In this paper, we construct a local-in-time mild solution in $L^1(\mathbb {R}^N)$ for $\phi \in X_q$ if $q\ge N/2$. We show that, for each $0\le q<N/2$, there is a nonnegative initial function $\phi _0\in X_q$ such that the problem has no nonnegative solution, using a necessary condition given by Baras–Pierre (Ann Inst Henri Poincaré Anal Non Linéaire 2:185–212, 1985). Since $X_q\subset X_{N/2}$ for $q\ge N/2$, $X_{N/2}$ becomes a sharp integrability condition. We also prove a uniqueness in a certain set of functions which guarantees the uniqueness of the solution constructed by our method.

1 Introduction and main results

We consider the existence and nonexistence for a Cauchy problem of the semilinear heat equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tu=\Delta u+|u|^{p-1}u &{} \text {in}\ \mathbb {R}^N\times (0,T),\\ u(x,0)=\phi (x) &{} \text {in}\ \mathbb {R}^N, \end{array}\right. } \end{aligned}$$

(1.1)

where $N\ge 1$, $p=1+2/N$ and $\phi$ is a possibly sign-changing initial function. When $\phi \in L^{\infty }(\mathbb {R}^N)$, one can easily construct a solution by using a fixed point argument. When $\phi \not \in L^{\infty }(\mathbb {R}^N)$, the solvability depends on the balance between the strength of the singularity of $\phi$ and the growth rate of the nonlinearity. Weissler [13] studied the solvability of (1.1), and obtained the following:

Proposition 1.1

Let $q_c:=N(p-1)/2$. Then, the following (i) and (ii) hold:

• (i) (Existence, subcritical and critical cases) Assume either both $q>q_c$ and $q\ge 1$ or $q=q_c>1$. The problem (1.1) has a local-in-time solution for $\phi \in L^q(\mathbb {R}^N)$.

• (ii) (Nonexistence, supercritical case) For each $1\le q<q_c$, there is $\phi \in L^q(\mathbb {R}^N)$ such that (1.1) has no local-in-time nonnegative solution.

Let u(x, t) be a function such that u satisfies the equation in (1.1). We consider the scaled function $u_{\lambda }(x,t):=\lambda ^{2/(p-1)}u(\lambda x,\lambda ^2t)$. Then, $u_{\lambda }$ also satisfies the same equation. We can easily see that $\left\| u_{\lambda }(x,0)\right\| _q=\left\| u(x,0)\right\| _q$ if and only if $q=q_c$. It is well known that $q_c$ is a threshold as Proposition 1.1 shows. However, the case $q=q_c=1$, i.e., $p=1+2/N$, is not covered by Proposition 1.1, and it is known that there is a nonnegative initial function $\phi \in L^1(\mathbb {R}^N)$ such that (1.1) with $p=1+2/N$ has no local-in-time nonnegative solution. See Brezis–Cazenave [2, Theorem 11], Celik–Zhou [3, Theorem 4.1] or Laister et al. [7, Corollary 4.5] for nonexistence results. See [1, 6, 11] and references therein for existence and nonexistence results with measures as initial data. In [2, Section 7.5], the case $p=1+2/N$ is referred to as “doubly critical case.” Several open problems were given in [2]. It was mentioned in [14, p.32] that (1.1) has a local-in-time solution if $\phi \in L^1(\mathbb {R}^N)\cap L^q(\mathbb {R}^N)$ for some $q>1$. However, a solvability condition was not well studied. See Table 1. For a detailed history about the existence, nonexistence and uniqueness of (1.1), see [3, Section 1].

Table 1 Existence and nonexistence of a local-in-time solution of (1.1) in $L^q(\mathbb {R}^N)$

In this paper, we obtain a sharp integrability condition on $\phi \in L^1(\mathbb {R}^N)$ which determines the existence and nonexistence of a local-in-time solution in the case $p=1+2/N$. We also show that a solution constructed in Theorem 1.3 is unique in a certain set of functions. Throughout the present paper, we define $f(u):=|u|^{p-1}u$. Let $L^q(\mathbb {R}^N)$, $1\le q\le \infty$, denote the usual Lebesgue space on $\mathbb {R}^N$ equipped with the norm $\left\| \,\cdot \,\right\| _q$. For $\phi \in L^1(\mathbb {R}^N)$, we define

$$\begin{aligned} S(t)[\phi ](x):=\int _{\mathbb {R}^N}G_t(x-y)\phi (y)\mathrm{d}y, \end{aligned}$$

where $G_t(x-y):=(4\pi t)^{-{N}/{2}}\exp \left( -\frac{|x-y|^2}{4t}\right)$. The function $S(t)[\phi ]$ is a solution of the linear heat equation with initial function $\phi$. We give a definition of a solution of (1.1).

Definition 1.2

Let u and ${\bar{u}}$ be measurable functions on $\mathbb {R}^N\times (0,T)$.

• (i) (Integral solution) We call u an integral solution of (1.1) if there is $T>0$ such that u satisfies the integral equation

  $$\begin{aligned} u(t)= & {} {\mathcal {F}}[u](t)\ \ \text {a.e.}\ x\in \mathbb {R}^N,\ \ 0<t<T,\ \ \text {and}\ \ \nonumber \\&\left\| u(t)\right\| _{\infty }<\infty \ \text {for}\ 0<t<T, \end{aligned}$$

  (1.2)

  where

  $$\begin{aligned} {\mathcal {F}}[u](t):=S(t)\phi +\int _0^tS(t-s)f(u(s))\mathrm{d}s. \end{aligned}$$

• (ii) (Mild solution) We call u a mild solution if u is an integral solution and $u(t)\in C([0,T),L^1(\mathbb {R}^N))$.

• (iii) We call ${\bar{u}}$ a supersolution of (1.1) if ${\bar{u}}$ satisfies the integral inequality ${\mathcal {F}}[{\bar{u}}](t)\le {\bar{u}}(t)<\infty$ for a.e. $x\in \mathbb {R}^N$, $0<t<T$.

For $0\le q<\infty$, we define a set of functions by

$$\begin{aligned} X_q:=\left\{ \phi (x)\in L^1_\mathrm{{loc}}(\mathbb {R}^N)\ \left| \ \int _{\mathbb {R}^N}|\phi |\left[ \log (e+|\phi |)\right] ^q\mathrm{d}x<\infty \right. \right\} . \end{aligned}$$

It is clear that $X_q\subset L^1(\mathbb {R}^N)$ and that $X_{q_1}\subset X_{q_2}$ if $q_1\ge q_2$. The main theorem of the paper is the following:

Theorem 1.3

Let $N\ge 1$ and $p=1+2/N$. Then, the following (i) and (ii) hold:

• (i) (Existence) If $\phi \in X_q$ for some $q\ge N/2$, then (1.1) has a local-in-time mild solution u(t), and this mild solution satisfies the following:

  $$\begin{aligned} \text {there is }C>0\text { such that }\left\| u(t)\right\| _{\infty }\le Ct^{-\frac{N}{2}}(-\log t)^{-q}\text { for small }t>0. \end{aligned}$$

  (1.3)

  In particular, (1.1) has a local-in-time mild solution for every $\phi \in X_{N/2}$.

• (ii) (Nonexistence) For each $0\le q<N/2$, there is a nonnegative initial function $\phi _0\in X_q$, which is explicitly given by (4.1), such that (1.1) has no local-in-time nonnegative integral solution, and hence (1.1) has no local-in-time nonnegative mild solution.

Remark 1.4

• (i) The function $\phi$ in Theorem 1.3(i) is not necessarily nonnegative.

• (ii) Theorem 1.3 indicates that $X_{N/2}(\subset L^1(\mathbb {R}^N))$ is an optimal set of initial functions for the case $p=1+2/N$ and $X_{N/2}$ is slightly smaller than $L^1(\mathbb {R}^N)$. This situation is different from the case $p>1+2/N$, since (1.1) is always solvable in the scale critical space $L^{N(p-1)/2}$ for $p>1+2/N$ (Proposition 1.1 (i)).

• (iii) $L^1(\mathbb {R}^N)$ is larger than the optimal set for $p=1+2/N$. On the other hand, it follows from Proposition 1.1(i) that if $1<p<1+2/N$, then (1.1) has a solution for all $\phi \in L^1(\mathbb {R}^N)$. Therefore, $L^1(\mathbb {R}^N)$ is small enough for the case $1<p<1+2/N$.

• (iv) The function $\phi _0$ given in Theorem 1.3(ii) is modified from $\psi (x)$ given by (1.9). This function comes from Baras–Pierre [1], and Theorem 1.3(ii) is a rather easy consequence of [1, Proposition 3.2]. However, we include Theorem 1.3(ii) for a complete description of the borderline property of $X_{N/2}$.

• (v) Laister et al. [7] obtained a necessary and sufficient condition for the existence of a local-in-time nonnegative solution of

  $$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tu=\Delta u+h(u) &{} \text {in}\ \mathbb {R}^N\times (0,T),\\ u(x,0)=\phi (x)\ge 0 &{} \text {in}\ \mathbb {R}^N. \end{array}\right. } \end{aligned}$$

  (1.4)

  They showed that when $h(u)=u^{1+2/N}[\log (e+u)]^{-r}$, (1.4) has a local-in-time nonnegative solution for every nonnegative $\phi \in L^1(\mathbb {R}^N)$ if $1<r<\lambda p$, and (1.4) does not always have if $0\le r\le 1$. Here, $\lambda >0$ is a certain constant. Therefore, the optimal growth of h(u) for $L^1(\mathbb {R}^N)$ is slightly smaller than $u^{1+2/N}$.

• (vi) The exponent $p=1+2/N$, which is called Fujita exponent, also plays a key role in the study of global-in-time solutions. If $1<p\le 1+2/N$, then every nontrivial nonnegative solution of (1.1) blows up in a finite time. If $p>1+2/N$, then (1.1) has a global-in-time nonnegative solution. See Fujita [4]. In particular, in the case $p=1+2/N$ we cannot expect a global existence of a classical solution for small initial data.

The next theorem is about the uniqueness of the integral solution in a certain class.

Theorem 1.5

Let $N\ge 1$, $p=1+2/N$ and $q>N/2$. Then, an integral solution u(t) of (1.1) is unique in the set

$$\begin{aligned} \left\{ u(t)\in L^1(\mathbb {R}^N)\ \left| \ \sup _{0\le t\le T}t^{N/2}(-\log t)^q\left\| u(t)\right\| _{\infty }<\infty \right. \right\} . \end{aligned}$$

(1.5)

Therefore, a solution given by Theorem 1.3 is unique.

Remark 1.6

• (i) If there were a solution that does not satisfy (1.5), then the uniqueness fails. However, it seems to be an open problem.

• (ii) In the case $q=N/2$, the uniqueness under (1.5) is left open.

• (iii) For general p and q, the uniqueness of a solution of (1.1) is known in the set

  $$\begin{aligned} \left\{ u(t)\in L^q(\mathbb {R}^N)\ \left| \ \sup _{0\le t\le T}t^{\frac{N}{2}\left( \frac{1}{q}-\frac{1}{pq}\right) }\left\| u(t)\right\| _{pq}<\infty \right. \right\} . \end{aligned}$$

  See Haraux–Weissler [5] and [13]. For an unconditional uniqueness with a certain range of p and q, see [2, Theorem 4].

• (iv) The nonuniqueness in $L^q(\mathbb {R}^N)$ is also known for (1.1). For $p>1+2/N$ and $1\le q<N(p-1)/2<p+1$, see [5]. For $p=q=N/(N-2)$, see Ni–Sacks [8] and Terraneo [12].

Let us mention technical details. We assume that $\phi \in X_q$ for some $q\ge N/2$. Using a monotone method, we construct a nonnegative mild solution w(t) of

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tw=\Delta w+f(w) &{} \text {in}\ \mathbb {R}^N\times (0,T),\\ w(x,0)=|\phi (x)| &{} \text {in}\ \mathbb {R}^N. \end{array}\right. } \end{aligned}$$

(1.6)

We define g(u) by

$$\begin{aligned} g(u):=u\left[ \log (\rho +|u|)\right] ^q, \end{aligned}$$

(1.7)

where $\rho >1$ is chosen appropriately. We will see that if $\rho \ge e$, then g(u) is convex for $u\ge 0$ and g plays a crucial role in the construction of the solution of (1.6). In order to construct a nonnegative solution we use a method developed by Robinson–Sierżȩga [10] with the convex function g, which was also used in Hisa–Ishige [6]. We define a sequence of functions $(u_n)_{n=0}^{\infty }$ by

$$\begin{aligned} {\left\{ \begin{array}{ll} u_n(t)={\mathcal {F}}[u_{n-1}](t)\ \text {for}\ 0\le t<T &{} \text {if}\ n\ge 1,\\ u_0(t)=0. &{} {} \end{array}\right. } \end{aligned}$$

(1.8)

Then, we show that $-w(t)\le u_n(t)\le w(t)$ for $0\le t<T$. Since $|u_n(t)|\le w(t)$, we can extract a convergent subsequence in $C_\mathrm{{loc}}(\mathbb {R}^N\times (0,T))$, using a parabolic regularization, the dominated convergence theorem and a diagonal argument. The limit function becomes a mild solution of (1.1).

In the nonexistence part, we use a necessary condition for the existence of a nonnegative solution of (1.1) obtained by Baras–Pierre [1], which is stated in Proposition 2.2 in the present paper. Using their result, one can show that there is $c_0>0$ such that if $\phi (x)\ge c_0\psi (x)$ in a neighborhood of the origin, then (1.1) has no nonnegative integral solution. Here,

$$\begin{aligned} \psi (x):=|x|^{-N}\left( -\log |x|\right) ^{-\frac{N}{2}-1}\ \ \text {for}\ \ 0<|x|<1/e. \end{aligned}$$

(1.9)

See also [6]. For each $0\le q<N/2$, we will see that a modified function $\phi _0$, which is given by (4.1), belongs to $X_q$. We show that $\phi _0$ does not satisfy the necessary condition for the existence of an integral solution stated in Proposition 2.2. Hence, (1.1) with $\phi _0$ has no nonnegative solution for each $0\le q<N/2$.

This paper consists of five sections. In Sect. 2, we recall known results including a monotone method, a necessary condition on the existence for (1.1) and $L^p$-$L^q$-estimates. In Sect. 3, we prove Theorem 1.3(i). In Sect. 4, we prove Theorem 1.3(ii). In Sect. 5, we prove Theorem 1.5.

2 Preliminaries

First, we recall the monotonicity method.

Lemma 2.1

Let $0<T\le \infty$, and let f be a continuous nondecreasing function such that $f(0)\ge 0$. The problem (1.1) has a nonnegative integral solution for $0<t<T$ if and only if (1.1) has a nonnegative supersolution for $0<t<T$. Moreover, if a nonnegative supersolution ${\bar{u}}(t)$ exists, then the solution u(t) obtained in this lemma satisfies $0\le u(t)\le {\bar{u}}(t)$.

Proof

This lemma is well known. See [10, Theorem 2.1] for details. However, we briefly show the proof for readers’ convenience.

If (1.1) has an integral solution, then the solution is also a supersolution. Thus, it is enough to show that (1.1) has an integral solution if (1.1) has a supersolution. Let ${\bar{u}}$ be a supersolution for $0<t<T$. Let $u_1=S(t)\phi$. We define $u_n$, $n=2,3,\ldots$, by

$$\begin{aligned} u_n={\mathcal {F}}[u_{n-1}]. \end{aligned}$$

Then, we can show by induction that

$$\begin{aligned} 0\le u_1\le u_2\le \cdots \le u_n\le \cdots \le {\bar{u}}<\infty \ \ \text {a.e.}\ x\in \mathbb {R}^N,\ 0<t<T. \end{aligned}$$

This indicates that the limit $\lim _{n\rightarrow \infty }u_n(x,t)$ which is denoted by u(x, t) exists for almost all $x\in \mathbb {R}^N$ and $0<t<T$. By the monotone convergence theorem, we see that

$$\begin{aligned} \lim _{n\rightarrow \infty }{\mathcal {F}}[u_{n-1}]={\mathcal {F}}[u], \end{aligned}$$

and hence $u={\mathcal {F}}[u]$. Then, u is an integral solution of (1.1). It is clear that $0\le u(t)\le {\bar{u}}(t)$. $\square$

Baras–Pierre [1] studied necessary conditions for the existence of an integral solution in the case $p>1$. See also [6] for details of necessary conditions including Proposition 2.2. The following proposition is a variant of [1, Proposition 3.2].

Proposition 2.2

Let $N\ge 1$ and $p=1+2/N$. If u(t) is a nonnegative integral solution, i.e., u(t) satisfies (1.2) with a nonnegative initial function $\phi$ and some $T>0$, then there exists a constant $\gamma _0>0$ depending only on N and p such that

$$\begin{aligned} \int _{B(\tau )}\phi (x)\mathrm{d}x\le \gamma _0|\log \tau |^{-\frac{N}{2}} \ \ \text {for all}\ \ 0<\tau <T, \end{aligned}$$

(2.1)

where $B(\tau ):=\{x\in \mathbb {R}^N\ |\ |x|<\tau \}$.

Lemma 2.3

Let $q\ge 0$ be fixed, and let

$$\begin{aligned} X_{q,\rho }:=\left\{ \phi \in L^1(\mathbb {R}^N)\ \left| \ \int _{\mathbb {R}^N}|\phi |\left[ \log (\rho +|\phi |)\right] ^q\mathrm{d}x<\infty \right. \right\} . \end{aligned}$$

(2.2)

Then, $\phi \in X_{q,\rho }$ for all $\rho >1$ if and only if $\phi \in X_{q,\sigma }$ for some $\sigma >1$.

Proof

We consider only the case $q>0$. It is enough to show that $\phi \in X_{q,\rho }$ for all $\rho >1$ if $\phi \in X_{q,\sigma }$ for some $\sigma >1$. Let $\rho >1$ be fixed, and let $\xi (s):=\log (\rho +s)/(\log (\sigma +s))$. By L’Hospital’s rule, we see that $\lim _{s\rightarrow \infty }\xi (s)=\lim _{s\rightarrow \infty }(s+\sigma )/(s+\rho )=1$. Since $\xi (s)$ is bounded on each compact interval in $[0,\infty )$, we see that $\xi (s)$ is bounded in $[0,\infty )$, and hence there is $C>0$ such that $\log (\rho +s)\le C\log (\sigma +s)$ for $s\ge 0$. This inequality indicates that $\phi \in X_{q,\rho }$ if $\phi \in X_{q,\sigma }$. $\square$

Because of Lemma 2.1, we do not care about $\rho >1$ in (2.2). In particular, if $\phi \in X_q$, then $\left\| g(\phi )\right\| _1<\infty$ for every $\rho >1$.

Proposition 2.4

• (i) Let $N\ge 1$ and $1\le \alpha \le \beta \le \infty$. There is $C>0$ such that, for $\phi \in L^{\alpha }(\mathbb {R}^N)$,

  $$\begin{aligned} \left\| S(t)\phi \right\| _{\beta } \le {C}{t^{-\frac{N}{2}\left( \frac{1}{\alpha }-\frac{1}{\beta }\right) }} \left\| \phi \right\| _{\alpha } \ \ \text {for}\ \ t>0. \end{aligned}$$

• (ii) Let $N\ge 1$ and $1\le \alpha <\beta \le \infty$. Then, for each $\phi \in L^{\alpha }(\mathbb {R}^N)$ and $C_0>0$, there is $t_0=t_0(C_0,\phi )$ such that

  $$\begin{aligned} \left\| S(t)\phi \right\| _{\beta }\le C_0t^{-\frac{N}{2}\left( \frac{1}{\alpha }-\frac{1}{\beta }\right) } \ \ \text {for}\ \ 0<t<t_0. \end{aligned}$$

For Proposition 2.4(i) (resp. (ii)), see [9, Proposition 48.4] (resp. [2, Lemma 8]). Note that $C_0>0$ in (ii) can be chosen arbitrary small.

We collect various properties of g defined by (1.7).

Lemma 2.5

Let $q>0$ and let $g_1(s):=s[\log (\rho +s)]^{-q}$. Then, the following holds:

• (i) If $\rho >1$, then $g'(s)>0$ for $s>0$.

• (ii) If $\rho \ge e$, then $g''(s)>0$ for $s>0$.

• (iii) If $\rho \ge e$, then $g_1(s)\le g^{-1}(s)$ for $s\ge 0$.

• (iv) If $\rho >1$, then there is $C_1>0$ such that $g^{-1}(s)\le g_1(C_1s)$ for $s\ge 0$.

• (v) If $\rho >e^{q/(p-1)}$, then $g^{-1}(s)^p/s$ is nondecreasing for $s\ge 0$.

• (vi) If $\rho \ge e$, then, for $\phi \in L^1(\mathbb {R}^N)$,

  $$\begin{aligned} S(t)\phi \le g^{-1}(S(t)g(\phi ))\ \ \text {for}\ \ t\ge 0. \end{aligned}$$

Proof

By direct calculation, we have

$$\begin{aligned} g'(s)&=[\log (\rho +s)]^{q-1}\left\{ \log (\rho +s)+\frac{q s}{s+\rho }\right\} ,\\ g''(s)&=\frac{q[\log (s+\rho )]^{q-2}}{(s+\rho )^2}\left[ s\left\{ \log (\rho +s)+q-1\right\} +2\rho \log (\rho +s)\right] . \end{aligned}$$

Thus, (i) and (ii) hold.

(iii) Since $\rho \ge e$, we have

$$\begin{aligned} g(g_1(s))= & {} \frac{s}{[\log (\rho +s)]^q} \left[ \log \left( \rho +\frac{s}{[\log (\rho +s)]^q}\right) \right] ^q\nonumber \\\le & {} \frac{s}{[\log (\rho +s)]^q}[\log (\rho +s)]^q=s \end{aligned}$$

(2.3)

for $s\ge 0$. By (i), we see that $g^{-1}(s)$ exists and it is increasing. By (2.3), we see that $g_1(s)\le g^{-1}(s)$ for $s\ge 0$.

(iv) Let $\xi (s):=(g(g_1(s))/s)^{1/q}=\log (\rho +\frac{s}{[\log (\rho +s)]^q})/(\log (\rho +s))$. Then, for each compact interval $I\subset [0,\infty )$, there is $c>0$ such that $\xi (s)>c$ for $s\in I$. By L’Hospital’s rule, we have

$$\begin{aligned} \lim _{s\rightarrow \infty }\xi (s)=\lim _{s\rightarrow \infty } \frac{1+\frac{\rho }{s}}{1+\frac{\rho }{s}[\log (\rho +s)]^q}\left\{ 1-\frac{1}{1+\frac{\rho }{s}}\frac{q}{\log (\rho +s)}\right\} =1, \end{aligned}$$

and hence there is $c_0>0$ such that $\xi (s)\ge c_0$ for $s\ge 0$. Thus, $g^{-1}(c_0^qs)\le g_1(s)$ for $s\ge 0$. Then, the conclusion holds.

(v) By (i), we see that $g(\tau )$ is increasing. Let $s:=g(\tau )$. Then, ${g^{-1}(s)^p}/{s}=\tau ^{p-1}\left[ \log (\rho +\tau )\right] ^{-q}$. Since $\rho >e^{q/(p-1)}$, we have

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}\tau }\frac{\tau ^{p-1}}{[\log (\rho +\tau )]^q}= \frac{\tau ^{p-2}}{[\log (\rho +\tau )]^{q+1}}\left\{ (p-1)\log (\rho +\tau )-\frac{q\tau }{\rho +\tau }\right\} >0. \end{aligned}$$

Thus, $g^{-1}(s)^p/s$ is increasing for $s\ge 0$.

(vi) Because of (ii), g is convex. By Jensen’s inequality, we see that $g(S(t)\phi )\le S(t)g(\phi )$. Since $g^{-1}$ exists and $g^{-1}$ is increasing, the conclusion holds. The proof is complete. $\square$

3 Existence

Lemma 3.1

Let $N\ge 1$ and $p=1+2/N$. Assume that $\phi \ge 0$. If $\phi \in X_q$ for some $q\ge N/2$, then (1.1) has a local-in-time nonnegative mild solution u(t), and $\left\| u(t)\right\| _{\infty }\le Ct^{-N/2}(-\log t)^{-q}$ for small $t>0$.

Proof

First, we consider the case $q=N/2$. Let $\rho \ge \max \{e^{q/(p-1)},e\}$ be fixed. Let g be defined by (1.7). Here, $q=N/2$ and g satisfies Lemma 2.5. We define

$$\begin{aligned} {\bar{u}}(t):=2g^{-1}(S(t)g(\phi )). \end{aligned}$$

We show that ${\bar{u}}$ is a supersolution. By Lemma 2.5(vi), we have

$$\begin{aligned} S(t)\phi \le g^{-1}\left( S(t)g(\phi )\right) =\frac{{\bar{u}}(t)}{2}. \end{aligned}$$

(3.1)

Next, we have

$$\begin{aligned}&\int _0^tS(t-s)f({\bar{u}}(s))\mathrm{d}s\nonumber \\&\quad =2^p\int _0^tS(t-s) \left[ S(s)g(\phi )\frac{g^{-1}\left( S(s)g(\phi )\right) ^p}{S(s)g(\phi )}\right] \mathrm{d}s\nonumber \\&\quad \le 2^pS(t)g(\phi )\int _0^t\left\| \frac{g^{-1}\left( S(s)g(\phi )\right) ^p}{S(s)g(\phi )}\right\| _{\infty }\mathrm{d}s\nonumber \\&\quad \le 2^pg^{-1}\left( S(t)g(\phi )\right) \left\| \frac{S(t)g(\phi )}{g^{-1}\left( S(t)g(\phi )\right) }\right\| _{\infty } \int _0^t\left\| \frac{g^{-1}\left( S(s)g(\phi )\right) ^p}{S(s)g(\phi )}\right\| _{\infty }\mathrm{d}s. \end{aligned}$$

(3.2)

Since $g(\phi )\in L^1(\mathbb {R}^N)$, by Proposition 2.4(ii) we have

$$\begin{aligned} \left\| S(t)g(\phi )\right\| _{\infty }\le C_0t^{-N/2}. \end{aligned}$$

(3.3)

By Lemma 2.5(v), we see that $g^{-1}(u)^p/u$ is nondecreasing for $u\ge 0$. Using (3.3) and Lemma 2.5(iv), we have

$$\begin{aligned}&\left\| \frac{g^{-1}\left( S(s)g(\phi )\right) ^p}{S(s)g(\phi )}\right\| _{\infty } \le \frac{g^{-1}\left( \left\| S(s)g(\phi )\right\| _{\infty }\right) ^p}{\left\| S(s)g(\phi )\right\| _{\infty }}\nonumber \\&\le \frac{g^{-1}(C_0s^{-N/2})^p}{C_0s^{-N/2}} \le \frac{C_1^pC_0^{2/N}}{s\left[ \log \left( \rho +C_0C_1s^{-N/2}\right) \right] ^{pq}} \le \frac{C_0^{2/N}C_1'}{s(-\log s)^{pq}}\quad \end{aligned}$$

(3.4)

for $0<s<s_0(C_0)$, where $C_1'$ is a constant independent of $C_0$. Using Lemma 2.5(iii) and (3.3), we have

$$\begin{aligned}&\left\| \frac{S(t)g(\phi )}{g^{-1}\left( S(t)g(\phi )\right) }\right\| _{\infty }\le \left\| \frac{S(t)g(\phi )}{g_1(S(t)g(\phi ))}\right\| _{\infty } =\left\| \left[ \log (\rho +S(t)g(\phi ))\right] ^q\right\| _{\infty }\nonumber \\&\quad \le \left[ \log (\rho +\left\| S(t)g(\phi )\right\| _{\infty })\right] ^q \le \left[ \log (\rho +C_0t^{-N/2})\right] ^q \le C_2'(-\log t)^q \end{aligned}$$

(3.5)

for $0<t<t_0(C_0)$, where $g_1$ is defined in Lemma 2.5 and $C_2'$ is a constant independent of $C_0$. By (3.4) and (3.5) we have

$$\begin{aligned}&\left\| \frac{S(t)g(\phi )}{g^{-1}\left( S(t)g(\phi )\right) }\right\| _{\infty } \int _0^t\left\| \frac{g^{-1}\left( S(s)g(\phi )\right) ^p}{S(s)g(\phi )}\right\| _{\infty }\mathrm{d}s\nonumber \\&\quad \le C_0^{2/N}C_1'C_2'(-\log t)^q\int _0^t\frac{\hbox {d}s}{s(-\log s)^{pq}}\nonumber \\&\quad =C_0^{2/N}C_1'C_2'(-\log t)^q\frac{2}{N(-\log t)^q} =C_0^{2/N}C_1'C_2'\frac{2}{N} \end{aligned}$$

(3.6)

for $0<t<\min \{s_0(C_0),t_0(C_0)\}$. By Proposition 2.4(ii), we can take $C_0>0$ such that $2^{p+1}C_0^{2/N}C_1'C_2'/N<1$. By (3.1), (3.2) and (3.6), we have

$$\begin{aligned} {\mathcal {F}}[{\bar{u}}](t) =S(t)\phi +\int _0^tS(t-s)f({\bar{u}}(s))\hbox {d}s \le \frac{1}{2}{\bar{u}}(t)+\frac{1}{2}{\bar{u}}(t)={\bar{u}}(t) \end{aligned}$$

for small $t>0$. Thus, there is $T>0$ such that ${\mathcal {F}}[{\bar{u}}]\le {\bar{u}}$ for $0<t<T$, and hence ${\bar{u}}$ is a supersolution. By Lemma 2.1, we see that there is $T>0$ such that (1.1) has a solution for $0<t<T$, and u(t) is clearly nonnegative. Moreover,

$$\begin{aligned} 0\le u(t)\le {\bar{u}}(t)=2g^{-1}(S(t)g(\phi ))\le Ct^{-\frac{N}{2}}(-\log t)^{-q}, \end{aligned}$$

(3.7)

which is the estimate in the assertion. We show that $u(t)\in C([0,T),L^1(\mathbb {R}^N))$. Since $\left\| g^{-1}(u)\right\| _1\le C\left\| u\right\| _1$, by (3.6) and Proposition 2.4(i) we have

$$\begin{aligned}&\left\| u(t)-S(t)\phi \right\| _1 \le \left\| \int _0^tS(t-s)f({\bar{u}}(s))\mathrm{d}s\right\| _1 \le C_0^{2/N}C_1'C_2'\frac{2}{N}\left\| g^{-1}(S(t)g(\phi ))\right\| _1\nonumber \\&\quad \le C_0^{2/N}C_1'C_2'\frac{2}{N}C\left\| S(t)g(\phi )\right\| _1 \le C_0^{2/N}C_1'C_2'\frac{2}{N}C'\left\| g(\phi )\right\| _1 \end{aligned}$$

(3.8)

for small $t>0$, where $C'$ is independent of $C_0$. By Proposition 2.4(ii), we can take $C_0>0$ arbitrary small, and hence

$$\begin{aligned} \left\| u(t)-S(t)\phi \right\| _1 \rightarrow 0\ \ \text {as}\ \ t\downarrow 0. \end{aligned}$$

Since S(t) is a strongly continuous semigroup on $L^1(\mathbb {R}^N)$ (see e.g., [9, Section 48.2]), we have

$$\begin{aligned} \left\| u(t)-\phi \right\| _1\le \left\| u(t)-S(t)\phi \right\| _1+\left\| S(t)\phi -\phi \right\| _1 \rightarrow 0\ \ \text {as}\ \ t\downarrow 0. \end{aligned}$$

(3.9)

It follows from (3.2) and (3.6) that $\left\| \int _0^tS(t-s)f({\bar{u}}(s))\mathrm{d}s\right\| _1<\infty$ for $0<t<T$. We see that if $0<t<T$, then

$$\begin{aligned} \left\| u(t+h)-u(t)\right\| _1\rightarrow 0\ \ \text {as}\ \ h\rightarrow 0. \end{aligned}$$

(3.10)

By (3.9) and (3.10), we see that $u(t)\in C([0,T),L^1(\mathbb {R}^N))$. The proof of (i) is complete.

Next, we consider the case $q>N/2$. The argument is the same until (3.6). We have

$$\begin{aligned}&\left\| \frac{S(t)g(\phi )}{g^{-1}\left( S(t)g(\phi )\right) }\right\| _{\infty } \int _0^t\left\| \frac{g^{-1}\left( S(s)g(\phi )\right) ^p}{S(s)g(\phi )}\right\| _{\infty }\mathrm{d}s\nonumber \\&\quad \le C_0^{2/N}C_1'C_2'(-\log t)^q\int _0^t\frac{\hbox {d}s}{s(-\log s)^{pq}}\nonumber \\&\quad =\frac{C_1^{2/N}C_1'C_2'}{pq-1}(-\log t)^{1-\frac{2q}{N}} \end{aligned}$$

(3.11)

instead of (3.6). Since the RHS of (3.11) goes to 0 as $t\downarrow 0$, the rest of the proof is almost the same with obvious modifications. In particular, (3.7) holds even for $q>N/2$. We omit the details. $\square$

We consider (1.6), where $\phi$ is given in (1.1). By Lemma 3.1, we see that (1.6) has a local-in-time solution which is denoted by w(t). We consider the sequence $(u_n)_{n=0}^{\infty }$ defined by (1.8). Then, the following lemma says that $\left\| u_n(t)\right\| _{\infty }$ can be controlled by w(t).

Lemma 3.2

Let $u_n$ be as defined by (1.8), and let w be a solution of (1.6) on (0, T). Then,

$$\begin{aligned} -w(t)\le u_n(t)\le w(t)\quad \mathrm{for a.e.}\ x\in \mathbb {R}^N\quad \mathrm{and}\ 0<t<T. \end{aligned}$$

(3.12)

Proof

It is clear from the definitions of $u_0$ and w(t) that

$$\begin{aligned} u_0(t)\le w(t)\ \ \text {for}\ \ 0<t<T. \end{aligned}$$

We assume that $u_{n-1}(t)\le w(t)$ on (0, T). Then, we have

$$\begin{aligned} w(t)= & {} S(t)|\phi |+\int _0^tS(t-s)f(w(s))\mathrm{d}s\\&\quad \ge S(t)\phi +\int _0^tS(t-s)f(u_{n-1}(s))\mathrm{d}s\\&\quad =u_n(t), \end{aligned}$$

and hence $u_n(t)\le w(t)$ for $0<t<T$. Thus, by induction we see that, for $n\ge 0$,

$$\begin{aligned} u_n(t)\le w(t)\ \ \text {on}\ \ 0<t<T. \end{aligned}$$

(3.13)

It is clear that $u_0(t)\ge -w(t)$ for $0<t<T$. We assume that $u_{n-1}(t)\ge -w(t)$ on (0, T). Then, we have

$$\begin{aligned} u_n(t)= & {} S(t)\phi +\int _0^tS(t-s)f(u_{n-1}(s))\mathrm{d}s\\\ge & {} -S(t)|\phi |+\int _0^tS(t-s)f(-w(s))\mathrm{d}s =-w(t), \end{aligned}$$

and hence, $u_n(t)\ge -w(t)$ on (0, T). Thus, by induction we see that for $n\ge 0$,

$$\begin{aligned} -w(t)\le u_n(t)\ \ \text {on}\ \ 0<t<T. \end{aligned}$$

(3.14)

By (3.13) and (3.14), we see that (3.12) holds. $\square$

Proof of Theorem 1.3

(i) Let $(u_n)_{n=0}^{\infty }$ be defined by (1.8). Using an induction argument with a parabolic regularity theorem, we can show that, for each $n\ge 1$, $u_n\in C^{2,1}(\mathbb {R}^N\times (0,T))$ and $u_n$ satisfies the equation

$$\begin{aligned} \partial _tu_n=\Delta u_n+f(u_{n-1})\ \ \text {in}\ \ \mathbb {R}^N\times (0,T) \end{aligned}$$

in the classical sense. Let K be an arbitrary compact subset in $\mathbb {R}^N\times (0,T)$, and let $K_1$, $K_2$ be two compact sets such that $K\subset K_1\subset K_2\subset \mathbb {R}^N\times (0,T)$. Because of Lemma 3.2, $f(u_{n-1})$ is bounded in $C(K_2)$. By a parabolic regularity theorem, we see that $u_n$ is bounded in $C^{\gamma ,\gamma /2}(K_1)$. Using a parabolic regularity theorem again, we see that $u_{n+1}$ is bounded in $C^{2+\gamma ,1+\gamma /2}(K)$.

In the following, we use a diagonal argument to obtain a convergent subsequence in $\mathbb {R}^N\times (0,T)$. Let $Q_j:=\overline{\{x\in \mathbb {R}^N|\ |x|\le j\}}\times \left[ \frac{T}{j+2},\frac{(j+1)T}{j+2}\right]$. Since $(u_n)_{n=3}^{\infty }$ is bounded in $C^{2,1}(Q_1)$, by Ascoli–Arzerà theorem there is a subsequence $(u_{1,k})\subset (u_n)$ and $u_1^*\in C(Q_1)$ such that $u_{1,k}\rightarrow u_1^*$ in $C(Q_1)$ as $k\rightarrow \infty$. Since $(u_{1,k})_{k=1}^{\infty }$ is bounded in $C^{2,1}(Q_2)$, there is a subsequence $(u_{2,k})\subset (u_{1,n})$ and $u_2^*\in C(Q_2)$ such that $u_{2,k}\rightarrow u_2^*$ in $C(Q_2)$ as $k\rightarrow \infty$. Repeating this argument, we have a double sequence $(u_{j,k})$ and a sequence $(u_j^*)$ such that, for each $j\ge 1$, $u_{j,k}\rightarrow u_j^*$ in $C(Q_j)$ as $k\rightarrow \infty$. We still denote $u_{n,n}$ by $u_n$, i.e., $u_n:=u_{n,n}$. It is clear that $u_{j_1}^*\equiv u_{j_2}^*$ in $Q_{j_1}$ if $j_1\le j_2$. Since $\mathbb {R}^N\times (0,T)=\bigcup _{j=1}^{\infty }Q_j$, there is $u^*\in C(\mathbb {R}^N\times (0,T))$ such that $u_n\rightarrow u^*$ in C(K) as $n\rightarrow \infty$ for every compact set $K\subset \mathbb {R}^N\times (0,T)$. In particular,

$$\begin{aligned} u_n\rightarrow u^*\ \ \text {a.e. in}\ \ \mathbb {R}^N\times (0,T). \end{aligned}$$

(3.15)

Let w be a solution of (1.6). It follows from Lemma 3.2 that $|u_n(x,t)|\le w(x,t)$. Since

$$\begin{aligned} |G_t(x-y)u_n(y,t)|\le |G_t(x-y)w(y,t)|\ \ \text {for}\ \ y\in \mathbb {R}^N, \end{aligned}$$

and

$$\begin{aligned} G_t(x-y)w(y,t)\in L^1_y(\mathbb {R}^N), \end{aligned}$$

by the dominated convergence theorem we see that

$$\begin{aligned} \lim _{n\rightarrow \infty }S(t)u_n= & {} \lim _{n\rightarrow \infty }\int _{\mathbb {R}^N}G_t(s-y)u_n(y,t)\mathrm{d}y\nonumber \\= & {} \int _{\mathbb {R}^N}G_t(s-y)u^*(y,t)\mathrm{d}y=S(t)u^*. \end{aligned}$$

(3.16)

By (3.2) and (3.6), we see that if $T>0$ is small, then

$$\begin{aligned} \int _0^t\int _{\mathbb {R}^N}G_{t-s}(x-y)f(w(y,s))dyds\le Cg^{-1}(S(t)g(\phi ))<\infty \end{aligned}$$

for each $(x,t)\in \mathbb {R}^N\times (0,T)$, and hence $G_{t-s}(x-y)f(w(y,s))\in L^1_{(y,s)}(\mathbb {R}^N\times (0,T))$. Since

$$\begin{aligned}&|G_{t-s}(x-y)f(u_{n-1}(y,s))|\\&\quad \le |G_{t-s}(x-y)f(w(y,s))|\ \ \text {for a.e.}\ (y,s)\in \mathbb {R}^N\times (0,T) \end{aligned}$$

and

$$\begin{aligned} G_{t-s}(x-y)f(w(y,s))\in L^1_{(y,s)}(\mathbb {R}^N\times (0,T)), \end{aligned}$$

by the dominated convergence theorem we see that

$$\begin{aligned}&\lim _{n\rightarrow \infty }\int _0^tS(t-s)f(u_{n-1}(s))\mathrm{d}s =\lim _{n\rightarrow \infty }\int _0^t\int _{\mathbb {R}^N}G_{t-s}(x-y)f(u_{n-1}(y,s))dyds\nonumber \\&\quad =\int _0^t\int _{\mathbb {R}^N}G_{t-s}(x-y)f(u^*(y,s))dyds =\int _0^tS(t-s)f(u^*(s))\mathrm{d}s. \end{aligned}$$

(3.17)

Thus, we take a limit of $u_n={\mathcal {F}}[u_{n-1}]$. By (3.15), (3.16) and (3.17), we see that $u^*(t)={\mathcal {F}}[u^*](t)$ for $0<t<T$.

Since $|u_n|\le w$, we see that $|u^*|\le w$. Since $|u^*|\le w$ in $\mathbb {R}^N\times (0,T)$, by (3.8) and the arbitrariness of $C_0>0$ we have

$$\begin{aligned} \left\| u^*(t)-S(t)\phi \right\| _1= & {} \left\| \int _0^tS(t-s)f(u^*(s))\mathrm{d}s\right\| _1\\\le & {} \left\| \int _0^tS(t-s)f(w(s))\mathrm{d}s\right\| _1 \rightarrow 0\ \ \text {as}\ \ t\downarrow 0. \end{aligned}$$

Then, $\left\| u^*(t)-\phi \right\| _1\le \left\| u^*(t)-S(t)\phi \right\| _1+\left\| S(t)\phi -\phi \right\| _1\rightarrow 0$ as $t\downarrow 0$. Since $\left\| \int _0^tS(t-s)f(w(s))\right\| _1<\infty$ for $0<t<T$, we can show by a similar way to the proof of Lemma 3.1 that $u^*(t)\in C((0,T),L^1(\mathbb {R}^N))$. Thus, $u^*(t)\in C([0,T),L^1(\mathbb {R}^N))$, and hence $u^*(t)$ is a mild solution. Since $|u^*(t)|\le w(t)$, by Lemma 3.1 we have (1.3). The proof of (i) is complete. $\square$

4 Nonexistence

Let $0\le q<N/2$ be fixed. Then, there is $0<\varepsilon <N/2-q$. We define $\phi _0$ by

$$\begin{aligned} \phi _0(x):= {\left\{ \begin{array}{ll} |x|^{-N}\left( -\log |x|\right) ^{-\frac{N}{2}-1+\varepsilon } &{} \text {if}\ |x|<1/e,\\ 0 &{} \text {if}\ |x|\ge 1/e. \end{array}\right. } \end{aligned}$$

(4.1)

Lemma 4.1

Let $0\le q<N/2$, and let $\phi _0$ be defined by (4.1). Then, the following holds:

• (i) $\phi _0\in X_q(\subset L^1(\mathbb {R}^N))$.

• (ii) The function $\phi _0$ does not satisfy (2.1) for any $T>0$.

Proof

(i) We write $\phi _0(r)=r^{-N}\left( -\log r\right) ^{-N/2-1+\varepsilon }$ for $0<r<1/e$. Since $\log (e+s)\le 1+\log s$ for $s\ge 0$, we have

$$\begin{aligned} \log (e+|\phi _0|)\le 1-N\log r-\left( \frac{N}{2}+1-\varepsilon \right) \log (-\log r)\le -2N\log r \end{aligned}$$

(4.2)

for $0<r<1/e$. Let $B(\tau ):=\{x\in \mathbb {R}^N\ |\ |x|<\tau \}$. Using (4.2), we have

$$\begin{aligned}&\int _{B(1/e)}|\phi _0|\left[ \log (e+|\phi _0|)\right] ^qdx \le \omega _{N-1}\int _0^{1/e}\frac{(2N)^q(-\log r)^qr^{N-1}\mathrm{d}r}{r^N(-\log r)^{N/2+1-\varepsilon }}\nonumber \\&\quad \le (2N)^q\omega _{N-1}\int _0^{1/e}\frac{\hbox {d}r}{r\left( -\log r\right) ^{N/2+1-q-\varepsilon }} =\frac{(2N)^q\omega _{N-1}}{\frac{N}{2}-q-\varepsilon }<\infty , \end{aligned}$$

(4.3)

where $\omega _{N-1}$ denotes the area of the unit sphere $\mathbb {S}^{N-1}$ in $\mathbb {R}^N$. By (4.3), we see that $\phi _0\in X_q$.

(ii) Suppose the contrary, i.e., there exists $\gamma _0>0$ such that (2.1) holds. When $0<\tau <1/e$, we have

$$\begin{aligned} \int _{B(\tau )}\phi _0(x)\mathrm{d}x= & {} \omega _{N-1}\int _0^{\tau }\frac{\hbox {d}r}{r(-\log r)^{N/2+1-\varepsilon }}\\= & {} \frac{C}{(-\log \tau )^{N/2-\varepsilon }}, \end{aligned}$$

where $C>0$ is independent of $\tau$. Then,

$$\begin{aligned} \gamma _0 \ge \frac{\int _{B(\tau )}\phi _0(x)\mathrm{d}x}{(-\log \tau )^{-N/2}}\ge C(-\log \tau )^{\varepsilon }\rightarrow \infty \ \ \text {as}\ \ \tau \downarrow 0. \end{aligned}$$

which is a contradiction. Thus, the conclusion holds. $\square$

Proof of Theorem 1.3 (ii)

Let $0\le q<N/2$. It follows from Lemma 4.1(i) that $\phi _0\in X_q$. By Lemma 4.1(ii), we see that there does not exist $\gamma _0>0$ such that (2.1) holds. By Proposition 2.2, the problem (1.1) with $\phi _0$ has no nonnegative integral solution. $\square$

5 Uniqueness

Proof of Theorem 1.5

Let $q>N/2$. Suppose that (1.1) has two integral solutions u(t) and v(t). Using Young’s inequality and the inequality $\left\| u(t)\right\| _{\infty }\le Ct^{-N/2}(-\log t)^{-q}$, we have

$$\begin{aligned} \left\| u(t)-v(t)\right\| _1&\le \int _0^t\left\| G_{t-s}*\left\{ \left( p|u|^{p-1}+p|v|^{p-1}\right) (u-v)\right\} \right\| _1\mathrm{d}s\\&\le p\int _0^t\left\| G_{t-s}\right\| _1\left( \left\| u\right\| _{\infty }^{p-1}+\left\| v\right\| ^{p-1}_{\infty }\right) \mathrm{d}s\sup _{0\le s\le t}\left\| u(s)-v(s)\right\| _1\\&\le C\int _0^t\frac{\hbox {d}s}{\left\{ s^{N/2}(-\log s)^q\right\} ^{p-1}}\sup _{0\le s\le t}\left\| u(s)-v(s)\right\| _1. \end{aligned}$$

Since

$$\begin{aligned} \int _0^t{s^{-N(p-1)/2}(-\log s)^{-(p-1)q}}\mathrm{d}s=\frac{N(-\log t)^{1-2q/N}}{2q-N} \end{aligned}$$

and $1-2q/N<0$, we can choose $T>0$ such that $C\int _0^t{s^{-N(p-1)/2}(-\log s)^{-(p-1)q}}\mathrm{d}s<1/2$ for every $0\le t\le T$. Then, we have

$$\begin{aligned} \sup _{0\le t\le T}\left\| u(t)-v(t)\right\| _1\le \frac{1}{2}\sup _{0\le s\le T}\left\| u(s)-v(s)\right\| _1, \end{aligned}$$

which implies the uniqueness. $\square$