Blow-up versus global existence of solutions for reaction–diffusion equations on classes of Riemannian manifolds

Abstract

It is well known from the work of Bandle et al. (J Differ Equ 251:2143–2163, 2011) that the Fujita phenomenon for reaction–diffusion evolution equations with power nonlinearities does not occur on the hyperbolic space ${\mathbb {H}}^N$, thus marking a striking difference with the Euclidean situation. We show that, on classes of manifolds in which the bottom $\Lambda$ of the $L^2$ spectrum of $-\Delta$ is strictly positive (the hyperbolic space being thus included), a different version of the Fujita phenomenon occurs for other kinds of nonlinearities, in which the role of the critical Fujita exponent in the Euclidean case is taken by $\Lambda$. Such nonlinearities are time-independent, in contrast to the ones studied in Bandle et al. (2011). As a consequence of our results we show that, on a class of manifolds much larger than the case $M={\mathbb {H}}^N$ considered in Bandle et al. (2011), solutions to (1.1) with power nonlinearity $f(u)=u^p$, $p>1$, and corresponding to sufficiently small data, are global in time. Though qualitative similarities with similar problems in bounded, Euclidean domains can be seen in the results, the methods are significantly different because of noncompact setting dealt with.

1 Introduction

We investigate existence of global in time solutions, versus blow-up in finite time, to nonlinear reaction–diffusion problems of the following type:

$$\begin{aligned} {\left\{ \begin{array}{ll} \, u_t= \Delta u +\,f(u) &{}\, \text {in}\,\, M\times (0,T) \\ \,\; u =u_0\ge 0\,&{}\text {in}\,\, M\times \{0\}, \end{array}\right. } \end{aligned}$$

(1.1)

where $\Delta$ is the Laplace–Beltrami operator on a Riemannian manifold M, $T\in (0,\infty ]$ and $f:[0,+\infty )\rightarrow [0,+\infty )$ is e.g., a locally Lipschitz, increasing function. Further specification of the assumptions on the manifold M and on the nonlinearity f will be made later on. A crucial parameter, according to whose value the behaviour of solutions will change, will be the value of $f'(0)$ when f is differentiable in $x=0$. It will be usually required that $u_0\in C(M)\cap L^\infty (M)$ to ensure the existence of classical solutions at least up to a certain time.

The analogue of (1.1) in the Euclidean setting has a long history especially in the particularly important case of power nonlinearities, i.e., for the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \, u_t= \Delta u +\,|u|^{p-1}u &{} \,\text {in}\,\, {\mathbb {R}}^n\times (0,T) \\ \,\; u =u_0 &{}\,\text {in}\,\, {\mathbb {R}}^n\times \{0\}, \end{array}\right. } \end{aligned}$$

(1.2)

where it is assumed that $u_0\in L^\infty ({\mathbb {R}}^n)$. It has been shown by Fujita in [5], and in [12] and [14] for the critical case, that for problem (1.2) the following facts hold:

• a) If $1 < p\le p^*:=\frac{N+2}{N}$, (1.2) does not possess nontrivial global solutions.

• b) If $p > p^*$ solutions corresponding to data that are sufficiently small in a suitable sense are global in time.

It should be noticed that, by a generalization of a result of Kaplan [13], solutions corresponding to sufficiently large data blow up for any $p>1$. A complete account of results concerning blow-up and global existence of solutions to semilinear parabolic equations posed in ${\mathbb {R}}^n$ can be found, e.g., in [1, 4, 16, 23] and in references therein.

In the case of evolution equation posed on Riemannian manifolds the situation may change completely. In fact, an analogue of (1.2) has been studied in [2] in the important context of the hyperbolic space ${\mathbb {H}}^n$, namely on the simply connected manifold of constant sectional curvature equal to $-1$. It is shown there that for all $p>1$ sufficiently small initial data give rise to global in time solutions. A kind of Fujita phenomenon nontheless takes place but when a time dependent modification of (1.2) is taken into account. Namely, if the reaction term $u^p$ (considering nonnegative data, hence nonnegative solutions) is replaced by $e^{\alpha t}u^p$, $\alpha >0$ being a fixed parameter, then a Fujita-type phenomenon then takes place, the threshold value being $p^\sharp :=1+\frac{\alpha }{\Lambda }$ where $\Lambda :=(N-1)^2/4$ is the bottom of the $L^2$ spectrum of $-\Delta$ on ${\mathbb {H}}^n$. Informally, one might say that for this to hold the nonlinearity must be amplified exponentially as time grows, in fact it is also shown in [2] that if the exponential factor in time is replaced by a power of time the Fujita phenomenon still does not occur. See also [24] for a careful analysis of the critical case $p=p^\sharp$, in which the authors show, combining their results with the ones of [2], that in such case global in time solutions exist for all values of $\alpha$, thus marking a further difference with the Euclidean case. Further precise results on the lifespan of solutions that are not global are given in [25].

Recently, a number of results concerning blow-up and global existence for solutions of nonlinear parabolic equations with power-like reaction term and nonlinear, slow diffusion of porous medium type has also been obtained, on some classes of Riemannian manifolds, in [9,10,11, 17, 26]. The results for this class of equations show usually several differences with the case involving a linear diffusion, already in the Euclidean case, see [22].

Our goal here will be to present some result for different type of time independent nonlinearities in which a new kind of Fujita phenomenon takes place, in a wide class of manifolds that includes the hyperbolic space. We shall consider noncompact complete Riemannian manifold M of infinite volume, with dimension $N\ge 3$, and require that some further conditions on M, to be described below, hold. In our first result, Theorem 3.1, we require the following additional condition on M:

• M is stochastically complete, and $\lambda _1(M):= \inf {\text {spec}}(-\Delta )\,\,>0$, where ${\text {spec}}(-\Delta )$ is the $L^2$ spectrum of the Laplace–Beltrami operator $-\Delta$.

Under such assumptions, we prove nonexistence of global solutions for problem  (1.1)  e.g., if  f is convex and increasing,  $f'(0)>\lambda _1(M)$, and 1/f is integrable at infinity.

Stochastic completeness is a well-studied property of Riemannian manifolds, and it amounts to requiring that $T_t1=1$ for all $t>0$, or equivalently that

$$\begin{aligned} \int _M p(x,y,t) \, \mathrm{d}\mu (y)\,= \,1,\quad \text {for all}\,\,\, x\in M,\,\, t>0. \end{aligned}$$

(1.3)

where p(x, y, t) is the heat kernel of the manifold M and $\mu$ the Riemannian measure. See e.g., [6, 7] for a number of conditions on M ensuring that stochastic completeness holds. For example, it suffices that, for some $o\in M$, the function $r \mapsto \frac{ r}{\log {\mathcal {V}}(o,r)}$ is not integrable at infinity, where ${\mathcal {V}}(o,r)$ is the volume of the geodesic ball of radius r centered at o. Note that this is true in particular if ${\mathcal {V}}(o,r)\le C e^{ar^2}$ for suitable $C,a>0$. This allows e.g., sectional curvatures to tend to $-\infty$ at the infinity of M (namely when $\varrho (o,x)\rightarrow +\infty$, $\varrho$ being the Riemannian distance and o being fixed in M), at most quadratically, in a suitable precise sense, see also [8] for relations to nonlinear elliptic and parabolic equations on M. As for the assumption $\lambda _1(M)\,>0$ we comment that a well-known sufficient condition for this to hold is that $\mathrm {sec}\, \le c<0$, $\mathrm {sec}$ denoting sectional curvatures. Thus, the class of manifolds on which the above result works is large, as it includes e.g., all those manifolds whose sectional curvatures are pinched between two strictly negative constants, and in particular the hyperbolic space.

In our second result, Theorem 3.2, the additional assumption we require on M beside the previous ones is the following:

• the following Faber–Krahn inequality holds: for some $c>0$, for any non-empty relatively compact open subset $\Omega \subset M$,

  $$\begin{aligned} \lambda _1(\Omega )\ge \frac{c}{[\mu (\Omega )]^{\frac{2}{N}}}, \end{aligned}$$

  (1.4)

  where $\lambda _1(\Omega )$ is the first eingevalue of the Laplace operator on $\Omega$ completed with homogeneous Dirichlet boundary conditions, and $\mu$ is the Riemannian measure.

Under such assumptions, we show existence of global solutions for small data e.g., if f is differentiable at $x=0$ with $f'(0)< \lambda _1(M)$.

We refer e.g., to [7, Cor. 14.23, Cor. 15.17] for equivalent conditions for the Faber–Krahn inequality to hold. In particular, its validity is implied by on-diagonal bounds for the heat kernel in the form $p(t,x,x)\le c\,t^{-N/2}$ for all $x\in M$, $t>0$, or by the validity of the Sobolev (or the Nash) inequality on M, which is in turn satisfied e.g., when sec$\,\le 0$ on M, sec denoting sectional curvatures.

Observe that our blow-up result is similar in character to the well-known blow-up result for bounded domains of ${\mathbb {R}}^n$ (see, e.g., [1, Section 3.2]). However, the methods of proof exploited for bounded domains do not work on general Riemannian manifolds, thus our arguments are completely different. Indeed, in bounded domains the blow-up result is usually obtained by means of the Kaplan method (see [1, 13]), which makes use of the first eigenvalue and of the first eigenfunction of the Laplacian. In order to extend that argument to a general Riemannian manifold, it would be necessary to know precisely the behaviour at infinity of the positive solution $\phi$ to

$$\begin{aligned} \Delta \phi +\lambda _1(M) \phi = 0 \quad \text { in }\, M\, \end{aligned}$$

and in particular its integrability properties w.r.t. the Riemannian measure, which are not known in general, e.g., on the hyperbolic space ${\mathbb {H}}^n$ $\phi$ belongs just to $L^{2+\varepsilon }$ for all $\varepsilon >0$. The Kaplan method yields indeed partial results on the subclass of Cartan–Hadamard manifolds, i.e., simply connected Riemannian manifolds with nonpositive sectional curvatures, with Ricci curvature bounded from below, see [20] when $f(u)=u^p$, but the method can not be pushed to get the sharp threshold value $\lambda _1(M)$ provided in Theorem 3.1, 3.2, even in the special case of Cartan–Hadamard manifolds and, in fact, even on the special case $M={\mathbb {H}}^n$.

Let us mention that some blow-up results in bounded domains have also been established in [18, 19], for more general operators, by means of the method of sub– and supersolutions. Those results seem to be quite implicit in character, and they are based on the asymptotic behaviour for large times of solutions to the associated linear problem.

As a further comment, we mention that it is easy to show that, on a wide class of manifolds, characterized by the validity of the parabolic maximum principle, a sufficient condition for which is e.g., the very general condition (6.2) below, blow-up of solutions corresponding to large data occurs, provided the nonlinearity f is convex and 1/f is integrable at infinity. See the end of this section for some more detail and Sect. 6 for a concise proof.

Let us now give briefly some more precise detail on the conditions on the nonlinearities to be verified in order to prove our results. In the first one, Theorem 3.1, we shall assume that f is continuous. Besides, we assume that $f\ge h$ where h is increasing and convex in $[0,+\infty )$, that it satisfies $\int ^{+\infty }\frac{1}{h(s)}\,\mathrm{d}s\,<+\infty$, that $h(0)=0$ and finally that the condition $h'(0)>\lambda _1(M)$ holds. Of course, a sufficient condition for this to hold is that the conditions satisfied by h are satisfied by the nonlinearity f itself, as mentioned above. Then we show that all solutions blow up in finite time. In Theorem 3.2 we shall show that, for any locally Lipschitz nonlinearity f such that $f(x)\le \lambda x$ in a neighbourhood of $x=0$, with $\lambda \le \lambda _1(M)$, then sufficiently small data give rise to solutions existing for all times. Of course, if $f'(0)$ exists, a sufficient condition for the above condition on f to hold for some $\lambda <\lambda _1(M)$ is $f'(0)<\lambda _1(M)$, as mentioned above. The combination of the two results thus shows the version of the Fujita phenomenon we aim at.

It is important to mention that the above results provide, as immediate consequences, new results w.r.t. the ones proved in [2] for (1.1) even in the classical case $f(u)=u^p$. In fact, Theorem 3.2 shows in particular that solutions corresponding to sufficiently small data are global, on a much wider class of manifolds than the hyperbolic space considered in one of the main result of [2], see Corollary 3.3 for a precise statement. In particular the results holds e.g., on all those manifolds whose sectional curvatures are pinched between two strictly negative constants everywhere.

The paper is organized as follows. In Sect. 2 we collect some preliminary material on the class of manifolds considered and on the concept of solution. In Sect. 3 we state all our main results. Section 4 contains the proof of Theorem 3.1, whereas Theorem 3.2 is proved in Sect. 5. In Sect. 6 we complement our main results by considering a large class of manifolds, e.g., those ones in which the radial Ricci curvature does not diverge at infinity faster that $-cr^2$, where r is the Riemannian distance from a given pole $o\in M$. We give a concise proof of the fact that on such manifolds, if f is a locally Lipschitz, increasing, convex function such that 1/f is integrable at infinity, large data give rise to solutions blowing up in finite time.

2 Preliminaries

2.1 Heat semigroup on M

Let $\{e^{t\Delta }\}_{t\ge 0}$ be the heat semigroup of M, acting on $L^p(M)$ for all $p\in [1,+\infty ]$. It admits a (minimal) heat kernel, namely a function $p\in C^{\infty }(M\times M\times (0,+\infty ))$, $p>0$ in $M\times M\times (0,+\infty )$ such that

$$\begin{aligned} (e^{t\Delta } u_0)(x)=\int _M p(x,y,t)\,u_0(y) \, \mathrm{d}\mu (y), \quad x\in M,\,\, t>0, \end{aligned}$$

for any $u_0\in L^p(M)$. It is well known that

$$\begin{aligned} \int _M p(x,y,t) \, \mathrm{d}\mu (y)\,\le \,1,\quad \text {for all}\,\,\, x\in M,\,\, t>0. \end{aligned}$$

(2.1)

As recalled in the Introduction, we say that a manifold M is stochastically complete if the following condition holds:

$$\begin{aligned} \int _M p(x,y,t) \, \mathrm{d}\mu (y)\,= \,1,\quad \text {for all}\,\,\, x\in M,\,\, t>0. \end{aligned}$$

See the considerations and the references after (1.3) for sufficient conditions for this fact to hold.

Furthermore, it is known that if M is a noncompact Riemannian manifold, then (see [3, Corollary 1])

$$\begin{aligned} \lim _{t\rightarrow +\infty } \frac{\log p(x,y,t)}{t}\,=\, -\lambda _1(M)\quad \text {locally uniformly in}\,\,\, M\times M, \end{aligned}$$

(2.2)

where $\lambda _1(M)$ is the infimum of the $L^2$ spectrum of $-\Delta$. We also recall that from the Faber–Krahn inequality (1.4) it follows that (see [7, Cor. 15.17 (b)]), for some ${\bar{C}}>0$:

$$\begin{aligned} p(x,y,t)\,\le \, {\bar{C}}\,e^{-\lambda _1\,t}\quad \text {for any}\,\,\, x,y\in M, t\ge 1. \end{aligned}$$

(2.3)

2.2 On the concept of solution

We shall always deal with bounded initial data. Solutions will be meant in the classical sense. More precisely, setting $Q_T=M\times (0,T]$, we require that $u\in C^{2,1}(Q_T)\cap C(\overline{Q_T})\cap L^\infty (\overline{Q_T})$ and that (1.1) holds in the classical sense.

We shall use in the sequel two different concepts of solution. On the one hand a function $u\in C(M\times (0,\tau ])\cap L^{\infty }(M\times (0,\tau ])$, for every $\tau \in (0,T]$ is called a mild solution of problem (1.1) if

$$\begin{aligned} u(x,t)=(e^{t\Delta }u_0)(x) + \int _0^t \left( e^{(t-s)\Delta }f(u)\right) (x)\,\mathrm{d}s, \end{aligned}$$

(2.4)

for any $t\in [0,\tau ]$.

We notice that, by adapting the methods of [2, Prop. 2.1, Lemma 2.1], for bounded initial data $u_0$ and up to a time T such that u(t) is bounded for all $t\in [0,T)$ (blow-up might occur at some positive time), the two concepts of solutions coincide provided f is locally Lipschitz, as required in our main results. Hence we shall use them indifferently when needed.

3 Statements of main results

In this section, we state our results concerning solutions to problem (1.1). We say that a solution blows up in finite time, whenever there exists $\tau >0$ such that

$$\begin{aligned} \lim _{t\rightarrow \tau ^-}\Vert u(t)\Vert _{L^\infty (M)}=+\infty . \end{aligned}$$

Otherwise, if a solution $u(t)\in L^\infty (M)$ for all $t>0$, we say that it is global. Our first results involves nonexistence of global solutions. Notice that the assumptions on the auxiliary function h below entail that h is differentiable in $x=0$.

Theorem 3.1

Let M be a complete, non compact, stochastically complete Riemannian manifold with $\lambda _1(M)>0$. Let $u_0\in C(M)\cap L^\infty (M), u_0\ge 0, u_0\not \equiv 0 \text { in } M$. Let f be locally Lipschitz in $[0, +\infty )$. Assume that $f\ge h$ where h is increasing and convex in $[0,+\infty )$ and $h(0)=0$. Moreover, suppose that

$$\begin{aligned} \int ^{+\infty }\frac{1}{h(s)}\,\mathrm{d}s\,<+\infty , \end{aligned}$$

(3.1)

and finally that $h'(0)>\lambda _1(M)$. Then any solution to problem (1.1) blows up in finite time.

Notice that in the above Theorem the fact that h is assumed to be increasing and convex implies the existence of $h'(0)$.

Theorem 3.2

Let M be a complete, non compact, stochastically complete Riemannian manifold with $\lambda _1(M)>0$ and such that the Faber–Krahn inequality (1.4) holds. Assume also that f is increasing, locally Lipschitz and $f(0)=0$. Moreover, suppose that, for some $\delta >0$ and $0<\alpha \le \lambda _1(M)$,

$$\begin{aligned} f(x)\le \alpha x\quad \text { for all }\,\, x\in [0, \delta ]. \end{aligned}$$

(3.2)

Furthermore, assume that $u_0\in C(M)\cap L^\infty (M)\cap L^1(M), u_0\ge 0 \text { in } M$ is small enough. Then there exists a global solution to problem (1.1); in addition, $u\in L^{\infty }(M\times (0,+\infty ))$.

The smallness condition on $u_0$ in Theorem 3.2 can be precisely formulated. Indeed, our hypothesis is that

$$\begin{aligned} \Vert u_0\Vert _{L^{\infty }(M)}\,\le \,\delta \,e^{-\alpha }, \end{aligned}$$

(3.3)

and

$$\begin{aligned} \Vert u_0\Vert _{L^{1}(M)}\,\le \,\frac{\delta }{C_2}, \end{aligned}$$

(3.4)

where $C_2={\bar{C}}\,e^{-(\lambda _1-\alpha )}$, ${\bar{C}}$ being defined in (2.3), while $\alpha$ and $\delta$ are given by (3.2).

As a consequence, we can generalize one of the main results of [2] (see also [21]) to a class of manifolds much wider than ${\mathbb {H}}^n$.

Corollary 3.3

Let M be a complete, non compact, stochastically complete Riemannian manifold with $\lambda _1(M)>0$ and such that the Faber–Krahn inequality (1.4) holds. Assume $f(x)=x^p$ for all $x\ge 0$ with $p>1$. Assume also that $u_0\in C(M)\cap L^\infty (M)\cap L^1(M), u_0\ge 0 \text { in } M$ is small enough. Then there exists a global solution to problem (1.1); in addition, $u\in L^{\infty }(M\times (0,+\infty ))$.

Remark 3.4

• For any $p>1$, let

  $$\begin{aligned} f(u)={\left\{ \begin{array}{ll} \alpha u, &{} u\,\in [0, 1],\\ \alpha u^p &{} u\,\in (1, +\infty ). \end{array}\right. } \end{aligned}$$

  If $\alpha >\lambda _1(M)$, then by Theorem 3.1, the solution to problem (1.1) blows up in finite time for any nontrivial $u_0$. On the other hand, if $\alpha \le \lambda _1(M)$, then the solution exists globally in time, provided that $u_0$ is sufficiently small.

• Let $f(u)=e^{\beta u}-1$ with $\beta >0$. By Theorem 3.1, if $\beta >\lambda _1(M)$, then the solution to problem (1.1) blows up in finite time. On the contrary, if $\beta <\lambda _1(M)$, then condition (3.2) is satisfied with $\beta <\alpha \le \lambda _1(M)$. Therefore, by Theorem 3.2, the solution exists globally in time, whenever $u_0$ is sufficiently small.

By standard methods, on suitable manifolds and for a wide class of nonlinearities f, it is possible to show that whenever $u_0$ is large enough, blow-up of solutions occurs. We defer the discussion of this fact to Sect. 6.

4 Finite time blow-up for any initial datum

4.1 Two key estimates

Let us first prove a preliminary lemma.

Lemma 4.1

Let M be a complete, non-compact Riemannian manifold with $\lambda _1(M)>0$. Let $u_0\in C(M)\cap L^\infty (M), u_0\ge 0, u_0\not \equiv 0 \text { in } M$. Let $\varepsilon \in (0,\lambda _1(M))$. Then there exist $\Omega \subset M$, $t_0>0$, $C_1>0$ such that

$$\begin{aligned} (e^{t\Delta }u_0)(x) \ge \, C_1 e^{-[\lambda _1(M)+\varepsilon ]t}, \quad \text {for any}\,\,\, x\in \Omega ,\,\,\,t>t_0. \end{aligned}$$

(4.1)

Proof

Let $\Omega \subset M$ be such that $\mu (\Omega )<+\infty$, $\int _{\Omega } u_0\, \mathrm{d}\mu >0$. From (2.2), there exists $t_0>0$ such that, for every $x,y\in \Omega$,

$$\begin{aligned} p(x,y,t)\ge e^{-[\lambda _1(M)+\varepsilon ]t} \quad \text {for every}\, t>t_0. \end{aligned}$$

Hence

$$\begin{aligned} \begin{aligned} (e^{t\Delta }u_0)(x)&\ge \int _M p(x,y,t) u_0(y)\, \mathrm{d}\mu (y) \\&\ge e^{-[\lambda _1(M)+\varepsilon ]t}\, \int _{\Omega }u_0(y) \,\mathrm{d}\mu (y). \end{aligned} \end{aligned}$$

Consequently, we obtain (4.1) with $C_1:= \int _{\Omega }u_0(y) \,\mathrm{d}\mu (y)>0.$ $\square$

Let u be a mild solution of equation (1.1), so that it fulfills (2.4). Then, for any $x\in M$ and for any $T>0$, we define

$$\begin{aligned} \Phi _x^T(t)\equiv \Phi _x(t):=\int _M p(x,z,T-t)\, u(z,t)\, \mathrm{d}\mu (z)\,\quad \text { for any }\,\, t\in [0, T]. \end{aligned}$$

(4.2)

Observe that

$$\begin{aligned} \Phi _x(0)=\int _M p(x,z,T) \,u_0(z)\, \mathrm{d}\mu (z)\,= (e^{T\Delta }u_0)(x), \,\, x\in M. \end{aligned}$$

(4.3)

Suppose that $u_0\in L^{\infty }(M)$. Choose any

$$\begin{aligned} \delta >\Vert u_0\Vert _{L^{\infty }(M)}. \end{aligned}$$

(4.4)

From (4.4) and (2.1) we obtain that, for any $x\in M$,

$$\begin{aligned} \begin{aligned} \Phi _x(0)&=\int _M p(x,z,T) \,u_0(z)\, \mathrm{d}\mu (z)\\&\le \,\Vert u_0\Vert _{L^{\infty }(M)}\int _M p(x,z,T)\, \mathrm{d}\mu (z)\\&< \delta . \end{aligned} \end{aligned}$$

(4.5)

We now state the following lemma.

Lemma 4.2

Let $M, f, h, u_0$ be as in Theorem 3.1. Let $x\in M$ and $\Phi _x(t)$ be as in (4.2). Set $\alpha :=h'(0).$ Then

$$\begin{aligned} \Phi _x(0)\,\,\le \,\, C\,e^{-\alpha T}, \quad \text {for any}\,\,T\ge {\bar{t}}, \end{aligned}$$

(4.6)

for suitable ${\bar{t}}>0$ and $C>0$, depending on x.

Note that ${\bar{t}}$ and C are given by (4.18) and (4.27) below, respectively, with $\delta$ as in (4.4).

Proof

Let u be a solution to problem (1.1). So (2.4) holds; hence

$$\begin{aligned} u(z,t)=\int _M p(z,y,t) u_0(y)\,\mathrm{d}\mu (y) + \int _0^t\int _{M} p(z,y,t-s)f(u)\,\mathrm{d}\mu (y)\, ds. \end{aligned}$$

(4.7)

In the definition of $\Phi _x^T(t)\equiv \Phi _x(t)$ (see (4.2)) fix any

$$\begin{aligned} T>\frac{1}{\alpha }\left[ \log \delta -\log \Phi _x(0)\right] . \end{aligned}$$

(4.8)

We multiply (4.7) by $p(x,z,T-t)$ and integrate over M. Therefore, we get

$$\begin{aligned} \begin{aligned} \int _M p(x,z,T-t)\,u(z,t)\,\mathrm{d}\mu (z)&=\int _M\int _M p(z,y,t)\,u_0(y)\,p(x,z,T-t)\,\mathrm{d}\mu (z)\,\mathrm{d}\mu (y)\\&\quad + \int _0^t\int _{M}\int _M p(z,y,t-s)\,f(u)\,p(x,z,T-t)\,\mathrm{d}\mu (z)\mathrm{d}\mu (y)ds. \end{aligned} \end{aligned}$$

(4.9)

Now, due to (4.2), for all $t\in (0, T),$ equality (4.9) reads

$$\begin{aligned} \begin{aligned} \Phi _x(t)&=\int _M\int _M p(z,y,t)\,u_0(y)\,p(x,z,T-t)\,\mathrm{d}\mu (z)\,\mathrm{d}\mu (y) \\&\quad + \int _0^t\int _{M}\int _M p(z,y,t-s)\,f(u)\,p(x,z,T-t)\,\mathrm{d}\mu (z)\,\mathrm{d}\mu (y)\,\mathrm{d}s. \end{aligned} \end{aligned}$$

By (4.3), for all $t\in (0, T),$

$$\begin{aligned} \begin{aligned} \Phi _x(t)&= \int _M p(x,y,T)\,u_0(y)\,\mathrm{d}\mu (y)+\int _0^t\int _{M} f(u)\,p(x,y,T-s)\,\mathrm{d}\mu (y)\,\mathrm{d}s \\&= \Phi _x(0)+\int _0^t\int _{M} f(u)\,p(x,y,T-s)\,\mathrm{d}\mu (y)\,\mathrm{d}s. \end{aligned} \end{aligned}$$

Since $f\ge h$ in $[0, +\infty$),

$$\begin{aligned} \begin{aligned} \Phi _x'(t)&= \int _{M} f(u)\,p(x,y,T-t)\,\mathrm{d}\mu (y) \\&\ge \int _{M} h(u)\,p(x,y,T-t)\,\mathrm{d}\mu (y). \end{aligned} \end{aligned}$$

(4.10)

Since h is an increasing convex function, due to (1.3), by using Jensen inequality, we get

$$\begin{aligned} \int _M p(x,y,T-t)\,h\left( u(y,t)\right) \,\mathrm{d}\mu (y)\ge & {}\, h\left( \int _M p(x,y,T-t)\,u(y,t)\,\mathrm{d}\mu (y)\right) \nonumber \\ {}= \,& {} h(\Phi _x(t)). \end{aligned}$$

(4.11)

Combining together (4.10) and (4.11), we obtain

$$\begin{aligned} \Phi _x'(t)\,\ge \, h(\Phi _x(t))\quad \text {for all }\,\, t\in (0, T). \end{aligned}$$

(4.12)

Fix any $x\in M$. We first observe that (4.12) implies that $\Phi _x(t)$ is an increasing function w.r.t the time variable t, since

$$\begin{aligned} \Phi _x'(t)\,>\,0\quad \text {for any}\,\,t\in (0, T). \end{aligned}$$

(4.13)

Moreover, due to (4.4) and (4.5), by continuity of $t\mapsto \Phi _x(t)$, we can infer that there exists $t_1>0$ such that

$$\begin{aligned} \Phi _x(t)<\delta \quad \text {for all }\,\, t\in (0, t_1). \end{aligned}$$

(4.14)

Since h is convex, increasing in $[0, +\infty ), h(0)=0, h'(0)=\alpha$, then

$$\begin{aligned} h(s)\ge \alpha s\quad \text { for all }\,\, s\ge 0. \end{aligned}$$

(4.15)

Due to (4.15) and to (4.14), we get

$$\begin{aligned} {\left\{ \begin{array}{ll} \Phi _x'(t)\,\ge \, \alpha \, \Phi _x(t)&{} \quad \text {for any}\,\,t\in (0,t_1), \\ \Phi _x(0)\,<\,\delta &{}. \end{array}\right. } \end{aligned}$$

Let

$$\begin{aligned} {\bar{t}}:=\sup \{t>0:\, \Phi _x(t)<\delta \}. \end{aligned}$$

We claim that

$$\begin{aligned} 0<{\bar{t}}\,\le \, -\frac{1}{\alpha }\log (\Phi _x(0)) +\frac{\log \delta }{\alpha }. \end{aligned}$$

(4.16)

In order to show (4.16), consider the Cauchy problem

$$\begin{aligned} {\left\{ \begin{array}{ll} y'(t) = \alpha y(t),&{} t>0\\ y(0)=\Phi _x(0). \end{array}\right. } \end{aligned}$$

Clearly,

$$\begin{aligned} y(t)=\Phi _x(0) e^{\alpha t}, \quad t>0. \end{aligned}$$

Hence

$$\begin{aligned} y(\tau )=\delta \quad \text {whenever }\,\, \tau =\frac{1}{\alpha }\left[ \log (\delta )-\log (\Phi _x(0))\right] . \end{aligned}$$

Furthermore, note that, in view of (4.5) and (4.8),

$$\begin{aligned} 0<\tau <T. \end{aligned}$$

(4.17)

By comparison,

$$\begin{aligned} \Phi _x(t)\ge y(t)\quad \text {for all }\, t\in (0, T). \end{aligned}$$

Thus, we can infer that there exists ${\bar{t}}\in (0, \tau \,]$ such that

$$\begin{aligned} \Phi _x({\bar{t}})=\delta . \end{aligned}$$

(4.18)

In particular, from (4.17) it follows that ${\bar{t}}<T.$

Due to (4.13) and (4.18), we obtain that

$$\begin{aligned} \Phi _x(t)\,>\,\delta \quad \text {for any}\,\,\, {\bar{t}}<t <T. \end{aligned}$$

(4.19)

By (4.12), in particular we have

$$\begin{aligned} \Phi _x'(t)\,\ge \,h(\Phi _x(t))\quad \text {for any}\,\,\,\bar{t}<t<T. \end{aligned}$$

(4.20)

Define

$$\begin{aligned} G(t):=\int _{\Phi _x(t)}^{+\infty } \frac{1}{h(z)}\,\mathrm{d}z\,\quad \text {for all }\,\, {\bar{t}}<t <T . \end{aligned}$$

Note that G is well-defined thanks to hypothesis (3.1) and to (4.19). Furthermore,

$$\begin{aligned} G'(t)=-\frac{\Phi _x'(t)}{h(\Phi _x(t))}\,\quad \text {for any }\,\, {\bar{t}}<t<T. \end{aligned}$$

(4.21)

We now define

$$\begin{aligned} w(t):=\exp \{G(t)\}\quad \text {for any}\,\,\,\bar{t}<t<T. \end{aligned}$$

(4.22)

Then, due to (4.21) and (4.20),

$$\begin{aligned} \begin{aligned} w'(t)&=-\frac{\Phi _x'(t)}{h(\Phi _x(t))}w(t)\\&\le - \,\,w(t)\,\quad \quad \quad \text {for any}\,\,\,\bar{t}<t<T. \end{aligned} \end{aligned}$$

(4.23)

By integrating (4.23) we get:

$$\begin{aligned} \begin{aligned} w(t)&\le w({\bar{t}}\,)\exp \left\{ -\int _{{\bar{t}}}^t\,\mathrm{d}s\right\} \\&\le w({\bar{t}}\,)\exp \left\{ -\,(t-{\bar{t}})\right\} \quad \text {for any}\,\,\,\bar{t}<t<T. \end{aligned} \end{aligned}$$

(4.24)

We substitute (4.22) into (4.24), so we have

$$\begin{aligned} \exp \{G(t)\}\,\le \,\exp \left\{ G({\bar{t}})-\,(t-{\bar{t}})\right\} \quad \text {for any}\,\,\,\bar{t}<t<T. \end{aligned}$$

Thus, for any ${\bar{t}}<t<T$,

$$\begin{aligned} G(t)\,\le \,G({\bar{t}})-\,(t-{\bar{t}}\,). \end{aligned}$$

(4.25)

We now combine (4.25) together with (4.16), hence

$$\begin{aligned} 0\le G(t)\,\le \,G({\bar{t}})-\,\left( t +\frac{1}{\alpha }\log (\Phi _x(0)) -\frac{\log \delta }{\alpha }\right) \quad \text {for any} \,\,\, {\bar{t}}<t<T. \end{aligned}$$

Hence

$$\begin{aligned} \log (\Phi _x(0))\,\le \, \alpha G({\bar{t}})+\log \delta -\alpha \,\,t\, \quad \text {for any} \,\,\, {\bar{t}}<t<T. \end{aligned}$$

(4.26)

We now take the exponential of both sides of (4.26). Thus we get, for any ${\bar{t}}<t<T$,

$$\begin{aligned} \begin{aligned} \Phi _x(0)&\le \exp \left\{ \alpha G({\bar{t}})+\log \delta -\alpha \,\,t\right\} \\&= C\,\exp \{-\alpha \,t\}, \end{aligned} \end{aligned}$$

where

$$\begin{aligned} C:=\exp \{\alpha G({\bar{t}})+\log \delta \}. \end{aligned}$$

(4.27)

This is the inequality (4.6). $\square$

4.2 Proof of Theorem 3.1

Proof

Take any $\delta >0$ fulfilling (4.4). We suppose, by contradiction, that u is a global solution of problem (1.1). Since $\alpha :=h'(0)>\lambda _1(M)$, there exists $\varepsilon \in (0,\alpha -\lambda _1(M))$ such that

$$\begin{aligned} \alpha >\lambda _1(M)+\varepsilon . \end{aligned}$$

Let $\Omega \subset M$ be such that

$$\begin{aligned} \mu (\Omega )<+\infty \quad \text {and}\quad \int _{\Omega } u_0 \, \mathrm{d}\mu >0. \end{aligned}$$

Then

$$\begin{aligned} \lim _{T\rightarrow +\infty } e^{\alpha -(\lambda _1(M)+\varepsilon )]T}=+\infty . \end{aligned}$$

(4.28)

Fix any arbitrary $x\in M$. By Lemmas 4.1 and 4.2,

$$\begin{aligned} C_1 \, e^{-[\lambda _1(M)+\varepsilon ]\,T}\,\,\le \,\,(e^{T\Delta }u_0)(x)\,\,\le \,\, C\,e^{-\alpha \, T}, \quad \text {for any}\,\,\, T>\max \{t_0, {\bar{t}}\,\}, \end{aligned}$$

where $t_0>0$, $C_1>0$ are given in Lemma 4.1, while ${\bar{t}}>0$, $C>0$ in Lemma 4.2. Hence, if u exists globally in time, we would have

$$\begin{aligned} e^{[\alpha -(\lambda _1(M)+\varepsilon )]\,T}\,\,\le \,\, \frac{C}{C_1} \quad \text {for any}\,\,\, T>\max \{t_0, {\bar{t}}\,\}. \end{aligned}$$

(4.29)

Nonetheless, due to (4.28), the left hand side of (4.29) tends to $+\infty$ as $T\rightarrow \infty$. Thus, we have a contradiction. Hence the thesis follows. $\square$

5 Global existence

Consider the linear Cauchy problem for the heat equation

$$\begin{aligned} {\left\{ \begin{array}{ll} v_t=\Delta v&{}\, \text {in}\,\,\, M\times (0,+\infty )\\ v=u_0&{}\, \text {in}\,\,\,M\times \{0\}, \end{array}\right. } \end{aligned}$$

(5.1)

with $u_0$ as in Theorem 3.2. Observe that problem (5.1) admits the classical solution

$$\begin{aligned} v(x,t)=\int _M p(x,y,t)\,u_0(y)\,\mathrm{d}y, \quad x\in M, t\ge 0. \end{aligned}$$

(5.2)

Hence, since $u_0\in L^\infty (M)$,

$$\begin{aligned} \Vert v(t)\Vert _{L^{\infty }(M)}\le \Vert u_0\Vert _{L^{\infty }(M)}\quad \text {for any}\,\,\,t>0. \end{aligned}$$

(5.3)

Moreover, since $u_0\in L^1(M)$, if the Faber–Krahn inequality holds, then, due to (2.3),

$$\begin{aligned} v(x,t) \le \bar{C}\, \Vert u_0\Vert _{L^1(M)}\,e^{-\lambda _1 t}\quad \text {for any}\,\,\,x\in M, t>1, \end{aligned}$$

(5.4)

where ${\bar{C}}$ has been defined in (2.3).

Let $\{\Omega _j\}_{j\in {\mathbb {N}}}\subset M$ be a sequence of domains such that

$$\begin{aligned} \begin{aligned}&\Omega _j\subset \Omega _{j+1}\quad \text {for any}\,\,\,j\in {\mathbb {N}},\\&\bigcup _{j\in {\mathbb {N}}} \Omega _j=M,\\&\partial \Omega _j\,\,\,\text {is smooth for every}\,\,\,j\in {\mathbb {N}}. \end{aligned} \end{aligned}$$

Furthermore, for every $j\in {\mathbb {N}}$ let $\zeta _j\in C_c^{\infty }(\Omega _j)$ be such that $0\le \zeta _j\le 1$, $\zeta _j\equiv 1$ in $\Omega _{j/2}$.

Proof of Theorem 3.2

We consider initial data $u_0$ satisfying (3.3) and (3.4). Define

$$\begin{aligned} \bar{u}(x,t):= e^{\alpha \,t}\,v(x,t),\quad x\in M, t\ge 0, \end{aligned}$$

with v and $\alpha$ given by (5.2) and (3.2), respectively.

Note that, due to (3.3) and (5.3), for any $x\in M, t\in (0,1]$,

$$\begin{aligned} 0\le {\bar{u}}(x,t)\le e^{\alpha \,t}\Vert v(t)\Vert _{L^{\infty }(M)}\,\le \, e^{\alpha \, t}\Vert u_0\Vert _{L^{\infty }(M)}\,\le \,\delta . \end{aligned}$$

(5.5)

Moreover, due to (3.4), (5.4), since $\alpha \le \lambda _1(M)$, for any $t>1$ we get

$$\begin{aligned} 0\le {\bar{u}}(x,t)\le e^{\alpha \,t}\Vert v(t)\Vert _{L^{\infty }(M)}\le \, \bar{C}\,\Vert u_0\Vert _{L^1(M)}e^{-(\lambda _1-\alpha )\, t}\le \,\delta . \end{aligned}$$

(5.6)

Inequalities (5.5) and (5.6) yield

$$\begin{aligned} 0\,\le \,\bar{u}(x,t)\,\le \, \delta \quad \text {for any}\,\,\,x\in M,\,\, t>0. \end{aligned}$$

(5.7)

Furthermore, we have

$$\begin{aligned} \bar{u}_t-\Delta \bar{u}-f({\bar{u}})=\alpha \,e^{\alpha \,t}\,v+e^{\alpha \,t}\,v_t-e^{\alpha \,t}\Delta v-f({\bar{u}}). \end{aligned}$$

Now, by using the fact that v is a classical solution to problem (5.1), due to (3.2) and (5.7), we get

$$\begin{aligned} \bar{u}_t-\Delta \bar{u}-f({\bar{u}})\ge \alpha {\bar{u}} - f({\bar{u}}) \ge \,0. \end{aligned}$$

(5.8)

Hence ${\bar{u}}$ is a weak supersolution to problem (1.1) in $M\times (0,\infty )$.

For any $j\in {\mathbb {N}}$ there exists a unique classical solution $u_j$ to problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u=\Delta u + f(u)&{} \quad \text {in}\,\,\,\Omega _j\times (0,T)\\ u=0&{} \quad \text {in}\,\,\,\partial \Omega _j\times (0,T)\\ u=\zeta _j\,u_0&{} \quad \text {in}\,\,\,\Omega _j\times \{0\}. \end{array}\right. } \end{aligned}$$

(5.9)

Clearly, $u_j\not \equiv 0$ because $u_0\,\zeta _j\not \equiv 0$ in $\Omega _j$. Moreover, for any $j\in {\mathbb {N}}$, in view of (5.8), since

$$\begin{aligned} v= u_0\ge \zeta _j u_0 \quad \text { in }\,\, M\times \{0\}, \end{aligned}$$

${\bar{u}}$ is a bounded weak supersolution of problem (5.9). Obviously, for any $j\in {\mathbb {N}}$, ${\underline{u}}\equiv 0$ is a subsolution to problem (5.9). Hence, by the comparison principle, for every $j\in {\mathbb {N}}$ we obtain

$$\begin{aligned} 0\,\le \,u_j\,\le \,\bar{u}\quad \text {for any}\,\,\,(x,t)\in \Omega _j\times (0,+\infty ). \end{aligned}$$

(5.10)

By standard a priori estimates (see, e.g., [15, Chapter 5]), we can infer that there exists a subsequence $\{u_{j_{k}}\}$ of $\{u_{j}\}$, which converges in $C^{2,1}_{x,t}(K\times [\varepsilon , T])$ as $k\rightarrow +\infty$, for each compact subset $K\subset M$ and for each $\varepsilon \in (0,T)$, and in $C_{loc}(M\times [0,T])$, to some function $u\in C^{2,1}_{x,t}(M\times (0,T])\cap C(M\times [0,T])$, which is a classical solution to problem (1.1). Moreover, from (5.10) we get

$$\begin{aligned} 0\le u\le \bar{u} \quad \text {in}\,\,\, M\times (0,+\infty ). \end{aligned}$$

Hence the thesis follows. $\square$

6 On blow-up of solutions for large data

In this section we discuss a blow-up result that can be obtained by standard tools. More precisely, we show that the solution to problem (1.1) blows up, provided that $u_0$ is large enough, and $f:[0, +\infty )\rightarrow [0, +\infty )$ is a locally Lipschitz, increasing, convex function fulfilling

$$\begin{aligned} \int ^{+\infty }\frac{1}{f(s)} \mathrm{d}s<+\infty . \end{aligned}$$

We need to introduce some preliminary material. Let $o\in M$ be a reference point and r(x) be the geodesic distance between x and o. For any $x\in M{\setminus }\{o\}$, denote by ${{\,\mathrm{Ric_o}\,}}$ the Ricci curvature at x in the radial direction $\frac{\partial }{\partial r}$. We assume that

$$\begin{aligned} {{\,\mathrm{Ric_o}\,}}(x) \ge - (N-1)\frac{\psi ''(r(x))}{\psi (r(x))}\quad \text { for all }\, x\in M{\setminus }\{o\}, \end{aligned}$$

for some $\psi \in C^\infty ((0, +\infty ))\cap C^1([0, +\infty ))$ such that $\psi '(0)=1, \psi (0)=0, \psi >0 \text { in } (0, +\infty )$ and

$$\begin{aligned} \int ^{+\infty }\frac{\int _0^r \psi ^{N-1}(\xi )\mathrm{d}\xi }{\psi ^{N-1}(r)}dr=+\infty . \end{aligned}$$

(6.1)

In view of such hypothesis, for problem (1.1) comparison principle for bounded sub- and supersolutions holds (see, e.g., [7, 21]). Condition (6.1) may be stated informally in a quite simpler way: a sufficient condition for this to hold is that

$$\begin{aligned} {{\,\mathrm{Ric_o}\,}}(x) \ge -c r(x)^2 \quad \text { for all }\, x\in M{\setminus }\{B_1(o)\} \end{aligned}$$

(6.2)

and a suitable $c>0$, as can be seen by choosing $\psi$ to be $e^{kr^2}$ in a neighborhood of infinity, for a suitable $k>0$.

Let D be an open precompact subset of M with smooth boundary. By Kaplan’s method (see [13]) it can be proved that, for some $v_0\in C({\bar{D}}), v_0\ge 0$ large enough, any solution v to problem

$$\begin{aligned} {\left\{ \begin{array}{ll} v_t= \Delta v +\,f(v) &{} \text {in}\,\, D\times (0,T) \\ v = 0 &{} \text {in}\,\, \partial D\times (0, T)\\ v =v_0 &{} \text {in}\,\, D\times \{0\}\, \end{array}\right. } \end{aligned}$$

(6.3)

blows up in finite time. Now, consider $u_0\in C(M), u_0\ge 0$ with compact support. Take any D as above containing the support of $u_0$ and set $v_0:=u_{0\lfloor {D}}$. By choosing $u_0$ big enough, and so $v_0$, the solution v to (6.3), corresponding to such $v_0$, blows up in a finite time, say $\tau >0$. Let

$$\begin{aligned} {\underline{u}}:={\left\{ \begin{array}{ll} v&{} \text {in } D\times (0, \tau )\\ 0&{} \text {in }(M{\setminus } D)\times (0, \tau ). \end{array}\right. } \end{aligned}$$

By the maximum principle,

$$\begin{aligned} v\ge 0 \quad \text { in }\,\, D\times (0, \tau ). \end{aligned}$$

Hence,

$$\begin{aligned} \frac{\partial v}{\partial n}\le 0 \quad \text { in }\,\, \partial D\times (0, \tau ), \end{aligned}$$

n being the outer unit normal vector to $\partial D$. This easily implies that, for any $0<T<\tau$, ${\underline{u}}$ is a bounded weak subsolution to problem (1.1). So, by comparison principle, for any solution u to problem (1.1),

$$\begin{aligned} u\ge {\underline{u}}\quad \text { in }\,\, M\times (0, T). \end{aligned}$$

Since v blows in finite time, the same holds for ${\bar{u}}$ and so for u.