Properties of solutions to porous medium problems with different sources and boundary conditions

Abstract

In this paper, we study nonnegative and classical solutions $u=u(\mathbf{x},t)$ to porous medium problems of the type

where $\Omega$ is a bounded and smooth domain of ${\mathbb {R}}^N$, with $N\ge 1$, $I=(0,t^*)$ is the maximal interval of existence of u, $m>1$ and $u_0(\mathbf{x})$ is a nonnegative and sufficiently regular function. The problem is equipped with different boundary conditions and depending on such boundary conditions as well as on the expression of the source g, global existence and blow-up criteria for solutions to ($\Diamond$) are established. Additionally, in the three-dimensional setting and when blow-up occurs, lower bounds for the blow-up time $t^*$ are also derived.

1 Introduction and motivations

It is well–known that several natural phenomena appearing in various physical, chemical and biological applications, are modeled through reaction diffusion equations. Their description, generally given in a cylinder $\Omega \times I$, where $\Omega$ is a bounded smooth domain of ${\mathbb {R}}^N$ ($N\ge 1$) with regular boundary $\partial \Omega$, and $I=(0,t^*)$, is formulated by an initial boundary value problem in the unknown $u=u(\mathbf{x},t)$ reading as

$$\begin{aligned} {\left\{ \begin{array}{ll} u_t=\nabla \cdot {\mathcal {A}}(u,\nabla u,\mathbf{x},t)+{\mathcal {B}}(u,\nabla u,\mathbf{x},t)&{} \mathbf{x} \in \Omega , t\in I,\\ u(\mathbf{x},0)=u_0(\mathbf{x})&{} \mathbf{x} \in \Omega ,\\ \text { Boundary conditions on }u &{} \mathbf{x} \in \partial \Omega , t\in I. \end{array}\right. } \end{aligned}$$

(1)

As to the question tied to the existence of local (i.e., $t^*$ finite) or global (i.e., $t^*=\infty$) solutions to classes of nonlinear problems of this type, sufficient conditions on ${\mathcal {A}}$ (as for instance, standard ellipticity behavior) as well as growth and regularity assumptions on both ${\mathcal {A}}$ and ${\mathcal {B}}$ guaranteeing this existence are known and have been widely studied in the literature (we refer, for instance, to [6, 20, 22, 23]).

In this paper, we dedicate our attention to problem (1) in the case ${\mathcal {A}}(u,\nabla u,\mathbf{x},t)=\nabla u^m$ and ${\mathcal {B}}(u,\nabla u,\mathbf{x},t)=g(u,|\nabla u|)$ and endowed with some boundary conditions, i.e.,

$$\begin{aligned} {\left\{ \begin{array}{ll} u_t= \Delta u^m+ g(u,|\nabla u|) &{} \mathbf{x}\in \Omega , t\in I, \\ k u_{\varvec{\nu }}+h u=0 &{} \mathbf{x}\in \partial \Omega , t \in I, \\ u(\mathbf{x},0)=u_0(\mathbf{x}) &{} \mathbf{x} \in \Omega , \end{array}\right. } \end{aligned}$$

(2)

where $\Omega$ and I were already introduced in the description of (1). Further, $\mathbf{\varvec{\nu }}=(\nu _1,\ldots ,\nu _N)$ stands for the outward normal unit vector to the boundary $\partial \Omega$, $\frac{\partial u}{\partial \varvec{\nu }}:=u_{ \varvec{\nu }}$ is the normal derivative of u, $m>1$, $k\ge 0$ and $h>0$. Additionally, $u_0:=u_0(\mathbf{x})\not \equiv 0$ is a nonnegative sufficiently smooth function (possibly also verifying compatible conditions on $\partial \Omega$), and $g(u,|\nabla u|)$ is a regular function of its arguments and is such that ${\underline{u}}\equiv 0$ represents a subsolution of the first equation in (2); henceforth, through the maximum principle, the nonnegativity on $\Omega \times I$ of solutions u to (2) remains essentially justified (see [23, 38]).

Beyond problems arising in the mathematical models for gas or fluid flow in porous media (see [5] and [43]), the formulation in (2) also describes the evolution of some biological population u occupying a certain domain whose growth is governed by the law of g (see [19]); precisely, the term $\Delta u^m$ idealizes the spread of the population, the parameter m indicating the speed of propagation: $m>1$ corresponds to slow, $0<m<1$ fast and the limit case $m=1$ infinity propagation. Moreover, when the coefficient k is zero (the well–known Dirichlet boundary conditions), then the distribution of u on the boundary of the domain maintains constant through the time, while for $k,h>0$, the Robin boundary conditions are recovered: they model a negative flux on the boundary, virtually meaning that the population u gets out of the domain with rate $-h/k$.

There are several investigations concerning different variants of the initial boundary value problem (2), all devoted to existence and properties of solutions: global and/or local existence, lower and upper bound of blow-up time, blow-up rates and/or asymptotic behavior. In our opinion, the following papers deserve to be referred also because they inspire this present work.

• Linear diffusion case ($m=1$) and $g(u,|\nabla u|)= u^p$, with $p>1$. For $\Omega ={\mathbb {R}}^N$, $N\ge 1$, in [4, 12] and [21] it is shown that for $1<p\le 1+(2/N)$ the problem has no global positive solution, while for $p>1+(2/N)$ it is possible to fix appropriate initial data $u_0$ emanating global solutions. When $\Omega$ is a bounded and smooth domain of ${\mathbb {R}}^3$ and Dirichlet boundary conditions are assigned, in [31] a lower bound for the blow-up time of solutions, if blow-up occurs, is derived, and [32] essentially deals with blow-up and global existence questions for the same problem in the N-dimensional setting, with $N\ge 2$, and endowed with Robin boundary conditions.

• Linear diffusion case ($m=1$) and $g(u,|\nabla u|)= k_1u^p-k_2|\nabla u|^q$, $k_1,k_2>0$ and $p,q\ge 1$. In [42] it is proved that for $q = 2p/(p+1)$ and small $k_2>0$ blow-up can occur for any $N \ge 1$, $p > 1$, $(N - 2)p < N + 2$ and without any restriction on the initial data, while lower bounds of the blow-up time, if blow-up occurs, are derived in [27] when $k_1$ and $k_2$ are time-dependent functions and under different boundary conditions.

• Linear diffusion case ($m=1$) and $g(u,|\nabla u|)= |\nabla u|^q$, with $q>2$. (The Hamilton–Jacobi equation.) In [36, 37], for certain bounded domains $\Omega$ of ${\mathbb {R}}^N$, $N\ge 1$, Dirichlet boundary conditions and regular data $u_0$, the authors discuss properties of solutions, known in the literature as gradient blow-up phenomena, loss of (classical) boundary conditions and recovery of boundary conditions. In particular, in the two-dimensional setting and for $2<q \le 3$, a sharp description of the final blow-up profile of $\nabla u$ near an isolated boundary singularity (in both normal and tangential directions) is given.

• Nonlinear diffusion case ($m>1$) and $g(u,|\nabla u|)=u^p$, with $p>1$. For $\Omega ={\mathbb {R}}^N$, $N\ge 1$, in [14, 15] and [25] it is shown that for $1<p\le m+(2/N)$ the problem has no global positive solution, while for $p>m+(2/N)$ there exist initial data $u_0$ emanating global solutions. When $\Omega$ is a bounded and smooth domain of ${\mathbb {R}}^N$, $N\ge 1$, and under Dirichlet boundary conditions, in [13] is proved that for $1<p<m$ the problem admits global solutions for all $u_0$ such that $u_0^{m-1}\in H_0^1(\Omega ),$ while for $m<p<m(1+(2/N))+(2/N)$ specific initial data produce unbounded solutions (see also [39]).

• Nonlinear diffusion case ($m>1$) and $g(u,|\nabla u|)=-u^p$, with $p>0$. The papers [16, 17] and [34] focus on results dealing with regularity and asymptotic behavior of solutions defined in the whole space ${\mathbb {R}}^N$, with $N\ge 1$. Additionally, for similar analysis in the case of bounded domains $\Omega$ of ${\mathbb {R}}^N$, $N\ge 1$, we also refer to [26].

• Nonlinear diffusion case ($m>1$) and $g(u,|\nabla u|)= u^p-u^\mu |\nabla u^\alpha |^q$, with $p,q,\alpha \ge 1$ and $\mu \ge 0$. With $\Omega$ bounded and smooth in ${\mathbb {R}}^N$, $N\ge 1$, and under Dirichlet boundary conditions, in [3] the authors treat the existence of the so-called admissible solutions and show that they are globally bounded if $p<\mu + mq$ or $m< p= \mu + mq$, as well as the existence of blowing-up admissible solutions, under the complementary condition $1\le \mu + mq < p$. Similarly, for $\alpha =m+\mu /q$, $m \ge 1, m/2 + \mu /q > 0, 1 \le q < 2$, existence of global weak solutions is addressed in [2].

In the context of this premise, we remark that our investigation is not focused on the question concerning the existence of solutions to system (2), but rather on their maximal interval of existence I. In particular, in the framework of nonnegative classical solutions, we follow the same approach used in largely cited papers (see, for instance, [30, 31, 33, 35, 40, 41] and references cited therein, for linear or nonlinear diffusion equations, even including our same case, i.e., systems like (2) with $m>1$) where such an existence is a priori assumed. Additionally, as to the lifespan I of these solutions, only two scenarios can appear and they provide the following extensibility criterion ( [6, 7, 20]):

$$\begin{aligned} \begin{aligned}&\triangleright )\; I=(0,\infty ), \hbox { so that } u \hbox { remains bounded for all } \mathbf{x}\in \Omega \hbox { and time } t>0, \\&\triangleright )\; I=(0,t^*), t^* \text { finite (the blow-up time), so that } \lim _{t \rightarrow t^*}||u(\cdot ,t)||_{L^\infty (\Omega )} =\infty . \end{aligned} \end{aligned}$$

(3)

By analyzing the expressions of g presented in the previous items, it is reasonable to expect that the contribution of the positive power addendum, representing a source which essentially increases the energy of the system, stimulates the occurrence of the blow-up; conversely, the negative terms have a damping effect, absorb the energy, and so, contrast the power source term.

Exactly in line with the state of the art above reviewed, with this paper, we aim at expanding the underpinning theory of the mathematical analysis for problem (2) when different choices of g, h and k are considered. Indeed, to the best of our knowledge, the interplay between both positive and negative powers of u, or $|\nabla u|$, in the source g and the Robin/Dirichlet boundary conditions has not yet been extensively studied. To be precise, our contribution includes blow-up and global existence criteria for nonnegative and classical solutions to (2) and estimates of the blow-up time when it occurs. We proof three theorems, summarized as follows:

• Criterion for blow-up in ${\mathbb {R}}^N$, $N\ge 1$: Theorem 3.1. If $g(u,|\nabla u|)=k_1u^p-k_2u^q$, $k_1,k_2,h>0$, $k=1$ and $p\ge \max \{m,q\}$ with $m,q> 1$, then the lifespan I of the nonnegative classical solution u to problem (2) emanating from any compatible initial data $u_0(\mathbf{x})$ complying with a certain largeness assumption, is finite and u blows up at some finite time $t^*$.

• Criterion for global existence in ${\mathbb {R}}^N$, $N\ge 1$: Theorem 3.2. If $g(u,|\nabla u|)=k_1u^p-k_2u^q$, $k_1,k_2,h>0$, $k=1$, $p<m$ with $m,q> 1$, then the lifespan I of the nonnegative classical solution u to problem (2) emanating from any initial data $u_0(\mathbf{x})$ is infinite and u is bounded for all time $t>0$.

• Lower bound of the blow-up time in ${\mathbb {R}}^3$: Theorem  3.3. If $g(u,|\nabla u|)=k_1u^p-k_2|\nabla u|^q$, $k_1,k_2,h>0$, $k=0$, for $2-1/p<m<p$ with $p> q \ge 2$, and u is a nonnegative classical solution to problem (2) emanating from any compatible initial data $u_0(\mathbf{x})$ and becoming unbounded in a certain measure at some finite time $t^{*}$, then, if $k_2$ is sufficiently large, there exists T such that $t^{*}\ge T$.

Remark 1

Even if the main motivation of this paper lies in enhancing the mathematical theory tied to nonlinear partial differential equations, we want to underline that the expressions of the function g given above are justified also by applicative reasons. Indeed, according to [42], a single (biological) species density u occupying a bounded portion of the space evolves in time by displacement, birth/reproduction and death. In particular, the births are described by a superlinear power of such a distribution, the natural deaths by a linear one and the accidental deaths by a function of its gradient; it leads to $u_t=\Delta u +C_1u^p-C_2u-C_3 |\nabla u|^q$, with $p,q>1$ and $C_1,C_2,C_3>0$. Adding to this equation homogeneous Dirichlet conditions corresponds to a non-viable environment on the boundary; homogeneous Neumann conditions stand for a totally insulated domain and Robin ones to a domain which allows the species to cross the boundary. Furthermore, other models originally introduced for only a single species describe the population growth through the preceding equation in which the source $C_1u^p-C_2u-C_3 |\nabla u|^q$ is replaced by the so-called logistic function $u(a-bu)$, with $a,b>0$, or more generally by functions independent of the accidental deaths and whose qualitative behavior is $u^l(1-u)$, with $l\ge 1$. All the mentioned sources have been also employed in chemotaxis models, precisely to describe the self-organizing of living organisms ([1, 8, 44,45,46,47,48]).

2 Main assumptions and preparatory lemmas

In this section, we give some crucial hypotheses, statements and lemmas which will be considered through the paper in the proofs of Theorems 3.2 and 3.3.

First we give these

Assumptions

For any integer $N\ge 1$ and real numbers $h>0,m>1$, we establish that:

(${\mathcal {H}}_{1}$):

$\Omega$ is a bounded domain of ${\mathbb {R}}^N$, star-shaped and convex in two orthogonal directions, whose geometry for some origin $\mathbf{x}_0$ inside $\Omega$ is defined throughout

$$\begin{aligned} m_1:= \frac{1}{ \min _{\partial \Omega } ((\mathbf{x}- \mathbf{x}_0) \cdot \mathbf{\varvec{\nu }} )}>0,\quad m_2:= \frac{\max _{{\overline{\Omega }}} | \mathbf{x}- \mathbf{x}_0|}{\min _{\partial \Omega } ((\mathbf{x}- \mathbf{x}_0) \cdot \mathbf{\varvec{\nu }})}>0. \end{aligned}$$

(4)

(${\mathcal {H}}_{2}$):

$\Omega$ is a bounded smooth domain of ${\mathbb {R}}^N$ such that

$$\begin{aligned} \frac{\xi _1(hm)}{hm} > m_1N+m_2, \end{aligned}$$

(5)

being $m_1$ and $m_2$ as in (4) and $\xi _1(h)$ the first positive eigenvalue associated with the supported membrane problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta w + \xi (h) w=0&{} \mathbf{x}\,\in \, \Omega , \\ w_{ \varvec{\nu }}+h w=0&{}\mathbf{x}\,\in \, \partial \Omega .\\ \end{array}\right. } \end{aligned}$$

(6)

Example 1

In order to provide triples $(h,m,\Omega )$ for which assumptions (${\mathcal {H}}_{1}$) and (${\mathcal {H}}_{2}$) are satisfied, let us fix N positive numbers $L_i>0$, $i=1,2,\ldots ,N$ and the N-dimensional rectangle-like domain $R^N_{\{L_1,L_2,\ldots ,L_N\}}(\mathbf{0})$, with center the origin and sizes $2L_i$. We have that

$$\begin{aligned} m_1=\frac{1}{\displaystyle \min _{\{i=1,2,\ldots ,N\}}L_i}, \quad m_2=m_1 \displaystyle \sqrt{\sum _{i=1}^NL_i^2} \end{aligned}$$

and, moreover, problem (6) can be explicitly solved by the separation of variables technique. Indeed, if we set $w (\mathbf{x})=w (x_1,x_2,\ldots ,x_N)=X_1(x_1)X_2(x_2)\ldots X_N(x_N)$, it is reduced to

$$\begin{aligned} {\left\{ \begin{array}{ll} -\displaystyle \sum _{i=1}^N \frac{X_i^{''}(x_i)}{X_i(x_i)}=\xi (h), \\ \pm X_i^{'}(\pm L_i)+hX_i(\pm L_i)=0, \quad \text {for all } i=1,2,\ldots ,N. \end{array}\right. } \end{aligned}$$

(7)

Evidently, this system is composed of N independent second-order ordinary differential problems, reading for each $i=1,2,\ldots ,N$ as

$$\begin{aligned} {\left\{ \begin{array}{ll} -\frac{X_i^{''}(x_i)}{X_i(x_i)}=\Lambda _i, \\ \pm X_i^{'}(\pm L_i)+hX_i(\pm L_i)=0, \end{array}\right. } \end{aligned}$$

(8)

where $\Lambda _i=\Lambda _i (h)>0$ is precisely the corresponding eigenvalue. ($\Lambda _i\le 0$ is not compatible with $h>0$ and $X_i\not \equiv 0$.) Subsequently, for some constants $C_1$ and $C_2$, we have that the general integral

$$\begin{aligned} X_i(x_i)=C_1\cos (\sqrt{\Lambda _i} x_i)+C_2\sin (\sqrt{\Lambda _i}x_i), \end{aligned}$$

must be such that

$$\begin{aligned} (h C_1\pm C_2\sqrt{\Lambda _i})\cos (\sqrt{\Lambda _i}L_i)+(h C_2\mp C_1\sqrt{\Lambda _i})\sin (\sqrt{\Lambda _i}L_i)=0. \end{aligned}$$

In order to ensure that the above system in the unknown $(C_1,C_2)$ admits a non–trivial solution, we impose its determinant equal to zero. It yields, for $z_i=L_i\sqrt{\Lambda _i}$

$$\begin{aligned} \bigg (h \cos (z_i) -\frac{z_i}{L_i}\sin (z_i)\bigg )\bigg (\frac{z_i}{L_i} \cos (z_i) +h\sin (z_i)\bigg )=0,\end{aligned}$$

or equivalently, dividing by $\cos (z_i)$,

$$\begin{aligned} \tan (z_i)=-\frac{z_i}{L_ih}\quad \text {or}\quad \tan (z_i)=\frac{h L_i}{z_i}. \end{aligned}$$

(9)

Since we are dealing with the smallest positive eigenvalue $\Lambda _i$ to (8), and it is seen that the first positive zero ${\hat{z}}_i$ of the first equation in (9) belongs to $\in (\pi /2,3 \pi /2)$, we focus on the second one, for which ${\hat{z}}_i\in (0,\pi /2).$ Finally, the same reasons apply for each independent problem given in (8), so that by superposition we conclude that $\xi (h)=\sum _{i=1}^N\sqrt{\Lambda _i}=\sum _{i=1}^N({\hat{z}}_i/L_i)^2$ is the first eigenvalue of (7). From this, we are now in the position to give an example complying with our assumptions (${\mathcal {H}}_{1}$) and (${\mathcal {H}}_{2}$) ; in fact, one can numerically check that the triple $(2,3/2,R^2_{\{\frac{2}{5},\frac{1}{7}\}}(\mathbf{0}))$ infers $\xi (hm) \cong 23.581$, and in turn $\xi (hm) - m h (2/L_1+\sqrt{L_1^2+L_2^2}/L_1)\cong 5.395$, so that it suitably fulfills (5).

The forthcoming two lemmas will be employed in the proof of Theorem 3.2 exactly in order to estimate a certain nonzero boundary integral when Robin boundary conditions are considered in system (2). In particular, Lemma 2.2 relies on assumptions (${\mathcal {H}}_{1}$) and (${\mathcal {H}}_{2}$), uses the result in Lemma 2.1 and even though it was already derived in [44, Lemma 3.3], for the sake of completeness we include its proof.

Lemma 2.1

Let $\Omega$ be a domain of ${\mathbb {R}}^N$ verifying assumption (${\mathcal {H}}_{1}$). For any nonnegative $C^1({\bar{\Omega }})$-function V, we have

$$\begin{aligned} \int \limits _{\partial \Omega } V^2 \mathrm{d} s \le m_1 N\int \limits _\Omega V^2 \mathrm{d} \mathbf{x}+2m_2 \int \limits _\Omega V|\nabla V|\mathrm{d} \mathbf{x}. \end{aligned}$$

(10)

Proof

This is relation [28, (A.1) of Lemma A.1.] with $n=1$ and written in terms of the coefficients in (4) of (${\mathcal {H}}_{1}$). $\square$

Lemma 2.2

Let $\Omega$ be a domain of ${\mathbb {R}}^N$ satisfying assumptions (${\mathcal {H}}_{1}$) and (${\mathcal {H}}_{2}$) . Then, for any nonnegative $C^1({\bar{\Omega }})$-function V verifying $V_{ \varvec{\nu }}+h V=0$ on $\partial \Omega$, we have

$$\begin{aligned} \int \limits _\Omega |\nabla V^m|^{2} \mathrm{d}{} \mathbf{x}\ge \sigma \int \limits _\Omega V^{2m}\mathrm{d}{} \mathbf{x}, \end{aligned}$$

(11)

with $\sigma :=\eta (hm)$, being $\eta (h)=(\xi _1(h)-h(m_1N+m_2))/(h m_2+1)>0$.

Proof

For any nonnegative $C^1({\bar{\Omega }})$-function V such that $V_{ \varvec{\nu }}+h V=0$ on $\partial \Omega$, the general Poincaré inequality returns this relation for the first eigenvalue $\xi _1(h)$ of (6):

$$\begin{aligned} \xi _1(h)\int \limits _\Omega V^2 \hbox {d}{} \mathbf{x} \le \int \limits _\Omega |\nabla V|^2\hbox {d}\mathbf{x}+h\int \limits _{\partial \Omega }V^2\hbox {d} s. \end{aligned}$$

It can be written, through relation (10) and subsequently the Young inequality with exponents 1 / 2, as

$$\begin{aligned} \xi _1(h)\int \limits _{\Omega } V^{2} \hbox {d}{} \mathbf{x} \le h(m_1N+m_2)\int \limits _{\Omega } V^{2}\hbox {d}{} \mathbf{x}+(hm_2+1)\int \limits _\Omega |\nabla V|^2 \hbox {d}{} \mathbf{x}. \end{aligned}$$

Hence, since $m_2>1$, we also have

$$\begin{aligned} \int \limits _{\Omega } |\nabla V|^2 \hbox {d}{} \mathbf{x} \ge \eta (h) \int \limits _\Omega V^2 \hbox {d}{} \mathbf{x}, \end{aligned}$$

(12)

with $\eta (h)=(\xi _1(h)-h(m_1N+m_2))/(h m_2+1).$ The function $\varphi =V^m$ verifies $\varphi _{ \varvec{\nu }}+hm \varphi =0$, so that (12) provides the

$$\begin{aligned} \int \limits _\Omega |\nabla V^m|^2 \hbox {d}{} \mathbf{x}=\int \limits _\Omega |\nabla \varphi |^2 \hbox {d}{} \mathbf{x} \ge \eta (hm) \int \limits _\Omega \varphi ^2 \hbox {d}{} \mathbf{x}=\sigma \int \limits _\Omega V^{2m} d\mathbf{x}, \end{aligned}$$

with $\sigma :=\eta (hm)=(\xi _1(hm)-hm(m_1N+m_2))/(hm m_2+1)$, that is positive by (5). $\square$

Similarly, the remaining two lemmas of this section are necessary to arrange terms emerging in the proof of Theorem 3.3, some of them also depending on $|\nabla u|$. The validity of these lemmas is strictly connected to some interplay between the parameters m, h, p, q defining our main problem, exactly as specified as follows:

Fixing parameters

Let $p> q \ge 2$. For $2-1/p<m<p$, we set

$$\begin{aligned} {\left\{ \begin{array}{ll} s=p-1, \quad \mu =\frac{q-1}{s}<1, \quad d=\frac{m-1}{s}<1,\\ 1< \delta < \frac{2}{3}(m+d)\frac{2m+3d-3}{2m+3d-1}, \quad \alpha = \frac{2(m+d)-\delta }{2(m+d)-3\delta }>1,\\ \beta =\frac{2m+3d-1}{2m+3d-3\alpha }>1, \quad \sigma =\frac{2(m+d)-3\delta }{2(m+d)}>0, \quad \gamma =d+\delta >1. \end{array}\right. } \end{aligned}$$

(13)

Lemma 2.3

Let $\Omega$ be a bounded domain of ${\mathbb {R}}^3$ with Lipschitz boundary, and let $m, \alpha , \beta , d, \sigma$ and $\gamma$ the constants defined in (13). Then for any nonnegative $C^1({\bar{\Omega }})$-function V

$$\begin{aligned} \int \limits _{\Omega } V^{m+1}\mathrm{d}{} \mathbf{x}\le \frac{\gamma - 1}{\gamma - \mu }\varepsilon _1 \int \limits _{\Omega } V^{m+\mu } \mathrm{d}{} \mathbf{x} +\frac{1 - \mu }{\gamma - \mu } \varepsilon _1^{-\frac{\gamma - 1}{1 - \mu }}\int \limits _{\Omega } V^{m+\gamma }\mathrm{d}{} \mathbf{x}, \end{aligned}$$

(14)

where $\varepsilon _1$ is an arbitrarily positive constant.

If, additionally, V vanishes on $\partial \Omega$, there exists a positive constant $C_S$ such that for every $\varepsilon _2>0$

$$\begin{aligned} \begin{aligned} \int \limits _\Omega V^{m+\gamma }\mathrm{d}{} \mathbf{x}&\le C_S^{\frac{3\delta }{m+d}+\frac{6\alpha }{2m+3d}}3\alpha d\frac{\sigma }{2m+3d} \varepsilon _2\int \limits _\Omega |\nabla V^{\frac{m+d}{2}}|^2 \mathrm{d}{} \mathbf{x} \\ {}&\quad +\,\Gamma ^{\frac{3\delta }{m+d}+\frac{6\alpha }{2m+3d}}d\sigma \frac{2m+3d-3\alpha }{2m+3d} \varepsilon _2\Big (\int \limits _\Omega V^m d\mathbf{x}\Big )^{\alpha \beta } \\ {}&\quad +\,(1-d)\varepsilon _2 C_S^\frac{3\delta }{m+d}\sigma \Big (\int \limits _\Omega V^m \mathrm{d}\mathbf{x}\Big )^{\alpha }\\ {}&quad+\,C_S^\frac{3\delta }{m+d}\frac{3\delta }{2(m+d)} \varepsilon _2^{1-\frac{2(m+d)}{3\delta }}\int \limits _\Omega |\nabla V^{\frac{m+d}{2}}|^2 \mathrm{d}{} \mathbf{x}. \end{aligned} \end{aligned}$$

(15)

Proof

Since $\mu <1$, for some positive constant $\gamma >1$, the Young inequality and the consideration of $\varepsilon _1>0$ yield

$$\begin{aligned} \begin{aligned} \int \limits _{\Omega } V^{m + 1}\hbox {d}{} \mathbf{x}&\le \Big ( \int \limits _{\Omega } V^{m + \mu }\hbox {d}{} \mathbf{x}\Big )^{\frac{\gamma - 1}{\gamma - \mu }}\Big ( \int \limits _{\Omega } V^{m+\gamma }\hbox {d}{} \mathbf{x}\Big )^{\frac{1 - \mu }{\gamma - \mu }} \\&\quad \frac{\gamma - 1}{\gamma - \mu }\varepsilon _1 \int \limits _{\Omega } V^{m+\mu } \hbox {d}{} \mathbf{x} +\frac{1 - \mu }{\gamma - \mu } \varepsilon _1^{-\frac{\gamma - 1}{1 - \mu }}\int \limits _{\Omega } V^{m+\gamma }\hbox {d}{} \mathbf{x}, \end{aligned} \end{aligned}$$

so that the first thesis is shown.

On the other hand, let V be such that $V=0$ on $\partial \Omega$: the Sobolev embedding in ${\mathbb {R}}^3$, $W_0^{1,2}(\Omega ) \hookrightarrow L^6(\Omega )$, provides a positive constant $C_S$ such that

$$\begin{aligned} \int \limits _\Omega \Big (V^\frac{m+d}{2}\Big )^6\hbox {d}{} \mathbf{x}\le C_S^6 \Big (\int \limits _\Omega |\nabla V^\frac{m+d}{2}|^2\hbox {d}{} \mathbf{x}\Big )^3. \end{aligned}$$

(16)

Now, for $\gamma =d+\delta >1$, the Hölder inequality leads to

$$\begin{aligned} \begin{aligned} \int \limits _{\Omega } V^{m + \gamma }\hbox {d}{} \mathbf{x}&=\int \limits _{\Omega } V^{(m + d)+\delta }\hbox {d}{} \mathbf{x} \\&\le \Big ( \int \limits _{\Omega } V^{m + d}\hbox {d}\mathbf{x}\Big )^{\frac{2(m+d)-\delta }{2(m+d)}}\Big ( \int \limits _{\Omega } (V^\frac{m+d}{2})^6\hbox {d}{} \mathbf{x}\Big )^{\frac{\delta }{2(m+d)}}, \end{aligned} \end{aligned}$$

(17)

so that by replacing (16) into (17), we obtain

$$\begin{aligned} \int \limits _{\Omega } V^{(m + d)+\delta }\hbox {d}{} \mathbf{x} \le C_S^{\frac{3\delta }{m+d}}\Big ( \int \limits _{\Omega } V^{m + d}\hbox {d}\mathbf{x}\Big )^{\frac{2(m+d)-\delta }{2(m+d)}}\Big (\int \limits _{\Omega } |\nabla V^\frac{m+d}{2}|^2\hbox {d}{} \mathbf{x}\Big )^{\frac{3\delta }{2(m+d)}}. \end{aligned}$$

The introduction of an arbitrary and positive constant $\varepsilon _2$, and an application of the Young inequality, allow us to write [recall (13)]

$$\begin{aligned} \begin{aligned} \int \limits _{\Omega } V^{(m + d)+\delta }\hbox {d}{} \mathbf{x}&\le C_S^{\frac{3\delta }{m+d}}\Big (\varepsilon _2\Big ( \int \limits _{\Omega } V^{m + d}\hbox {d}{} \mathbf{x}\Big )^{\frac{2(m+d)-\delta }{2(m+d)-3\delta }}\Big )^{\frac{2(m+d)-3\delta }{2(m+d)}} \\&\quad \times \, \Big (\varepsilon _2^{1-\frac{2(m+d)}{3\delta }} \int \limits _{\Omega } |\nabla V^\frac{m+d}{2}|^2\hbox {d}{} \mathbf{x}\Big )^\frac{3\delta }{2(m+d)} \\ {}&\le \, C_S^{\frac{3\delta }{m+d}}\varepsilon _2{\frac{2(m+d)-3\delta }{2(m+d)}}\Big ( \int \limits _{\Omega } V^{m + d}\hbox {d}\mathbf{x}\Big )^{\frac{2(m+d)-\delta }{2(m+d)-3\delta }}\\&\quad +\,C_S^{\frac{3\delta }{m+d}}\varepsilon _2^{1-\frac{2(m+d)}{3\delta }}\frac{3\delta }{2(m+d)} \int \limits _{\Omega } |\nabla V^\frac{m+d}{2}|^2\hbox {d}{} \mathbf{x}. \end{aligned} \end{aligned}$$

(18)

To bound the term $\Big (\int \limits _{\Omega } V^{m + d}\hbox {d}\mathbf{x}\Big )^{(2(m+d)-\delta )/(2(m+d)-3\delta )}$, let us observe that the Hölder and the Schwarz inequalities give, respectively,

$$\begin{aligned} \int \limits _\Omega V^{m+1}\hbox {d}{} \mathbf{x} \le \Big (\int \limits _\Omega V^{2(m+d)}\hbox {d}\mathbf{x}\Big )^\frac{1}{m+2d}\Big (\int \limits _\Omega V^m \hbox {d}\mathbf{x}\Big )^\frac{m+2d-1}{m+2d}, \end{aligned}$$

(19)

and

$$\begin{aligned} \int \limits _\Omega V^{2(m+d)}\hbox {d}{} \mathbf{x} \le \Big [\int \limits _\Omega \Big (V^\frac{m+d}{2}\Big )^6\hbox {d}{} \mathbf{x}\int \limits _\Omega V^{m+d}\hbox {d}\mathbf{x}\Big ]^\frac{1}{2}. \end{aligned}$$

(20)

Now, using in (20) relation (16), we get

$$\begin{aligned} \int \limits _\Omega V^{2(m+d)}\hbox {d}{} \mathbf{x} \le C_S^3 \Big (\int \limits _\Omega |\nabla V^{\frac{m+d}{2}}|^2\hbox {d}{} \mathbf{x}\Big )^\frac{3}{2}\Big (\int \limits _\Omega V^{m+d}\hbox {d}{} \mathbf{x}\Big )^\frac{1}{2}, \end{aligned}$$

and hence (19) reads

$$\begin{aligned} \begin{aligned} \int \limits _\Omega V^{m+1}\hbox {d}{} \mathbf{x}&\le C_S^\frac{3}{m+2d}\Big (\int \limits _\Omega |\nabla V^{\frac{m+d}{2}}|^2\hbox {d}\mathbf{x}\Big )^\frac{3}{2(m+2d)} \Big (\int \limits _\Omega V^{m+d}\hbox {d}\mathbf{x}\Big )^{\frac{1}{2(m+2d)}} \\ {}&\quad \times \, \Big (\int \limits _\Omega V^m \hbox {d}{} \mathbf{x}\Big )^{\frac{m+2d-1}{m+2d}}. \end{aligned} \end{aligned}$$

(21)

In addition, we first use again the Hölder inequality to lead to

$$\begin{aligned} \int \limits _\Omega V^{m+d}\hbox {d}{} \mathbf{x} \le \Big (\int \limits _\Omega V^{m+1}\hbox {d}\mathbf{x}\Big )^d \Big (\int \limits _\Omega V^m \hbox {d}{} \mathbf{x}\Big )^{1-d}, \end{aligned}$$

(22)

and then we insert this estimate in (21); combining terms, applying

$$\begin{aligned} a^r b^{1-r}\le r a + (1-r) b, \end{aligned}$$

(23)

valid for $a,b\ge 0$ and $0<r<1$, we arrive at ($\alpha$ as in (13))

$$\begin{aligned} \begin{aligned} \Big (\int \limits _\Omega V^{m+1}\hbox {d}{} \mathbf{x}\Big )^\alpha&\le C_S^{\frac{6\alpha }{2m+3d}}\Big (\int \limits _\Omega |\nabla V^\frac{m+d}{2}|^2\hbox {d}\mathbf{x}\Big )^\frac{3\alpha }{2m+3d}\Big (\int \limits _\Omega V^m \hbox {d}\mathbf{x}\Big )^{\alpha \frac{2m+3d-1}{2m+3d}}\\ {}&\le \, C_S^{\frac{6\alpha }{2m+3d}} \frac{3\alpha }{2m+3d} \int \limits _\Omega |\nabla V^\frac{m+d}{2}|^2\hbox {d}{} \mathbf{x}\\&\quad +\, C_S^{\frac{6\alpha }{2m+3d}} \frac{2m+3d-3\alpha }{2m+3d}\Big (\int \limits _\Omega V^m \hbox {d}\mathbf{x}\Big )^{\alpha \frac{2m+3d-1}{2m+3d-3\alpha }}. \end{aligned} \end{aligned}$$

(24)

Hence, by rearranging again (22) with (23) we attain

$$\begin{aligned} \begin{aligned} \Big (\int \limits _\Omega V^{m+d}\hbox {d}{} \mathbf{x}\Big )^\alpha&\le \, \Big [\Big (\int \limits _\Omega V^{m+1} \hbox {d}{} \mathbf{x} \Big )^d \Big (\int \limits _\Omega V^m \hbox {d}{} \mathbf{x}\Big )^{1-d}\Big ]^\alpha \\&\le \, d \Big (\int \limits _\Omega V^{m+1} \hbox {d}{} \mathbf{x} \Big )^\alpha +(1-d)\Big (\int \limits _\Omega V^m \hbox {d}{} \mathbf{x}\Big )^\alpha , \end{aligned} \end{aligned}$$

so that in view of (24) expression (18) (recall $\gamma =d+\delta$) infers our thesis. $\square$

Lemma 2.4

Let m, d and $\delta$ as in (13). If $c_1, c_2,\ldots ,c_6$ are positive real numbers satisfying

$$\begin{aligned} c_3 \ge c_5\Big (\frac{c_5}{c_6}\Big )^\frac{-3\delta }{2(m+d)} \Big (\frac{3\delta }{2m+2d-3\delta }\Big )^{-\frac{3\delta }{2(m+d)}} \frac{2(m+d)}{2(m+d)-3\delta }, \end{aligned}$$

(25)

then there exits $\xi _m \in (0,\infty )$ such that

$$\begin{aligned} c_5\xi _m+c_6\xi _{m}^{1-\frac{2(m+d)}{3\delta }}-c_3\le 0. \end{aligned}$$

(26)

Proof

For any $\xi \in (0,\infty )$, the function $\Phi (\xi ):=c_5\xi +c_6\xi ^{1-2(m+d)/3\delta }$ attains its minimum at the point

$$\begin{aligned} \xi _m=\Big (\frac{3\delta c_5}{c_6(2m+2d-3\delta )}\Big )^\frac{-3\delta }{2(m+d)}. \end{aligned}$$

(27)

Therefore, since (25) holds we have

$$\begin{aligned} c_3\ge c_5\Big (\frac{c_5}{c_6}\Big )^\frac{-3\delta }{2(m+d)}\Big (\frac{3\delta }{2m+2d-3\delta }\Big )^{-\frac{3\delta }{2(m+d)}}\frac{2(m+d)}{2(m+d)-3\delta }=\Phi (\xi _m),\end{aligned}$$

and relation (26) is proven. $\square$

3 Analysis and proofs of the main results

In this section, we discuss and give the demonstrations of our main theorems, whose general overview was summarized in Sect. 1.

3.1 A criterion for blow-up

The first theorem is dedicated to understand properties of solutions to system (2) when $g(u,|\nabla u|)=k_1u^p-k_2u^q$ and under Robin boundary conditions. Essentially, we observe that if the power q of the absorption term in g, as well as the coefficient m of the diffusion, does not surpass the power p from the growth contribution, then the occurrence of blow-up phenomena at some finite time may appear for some initial data $u_0(\mathbf{x})$, despite the outflow boundary conditions; in particular, no global solution is expected.

Theorem 3.1

Let $\Omega$ be a bounded smooth domain of ${\mathbb {R}}^N$, $N\ge 1,$ $k_1,k_2,h>0$, $k=1$, $p\ge \max \{m,q\}$, with $m,q > 1$, $g(u,|\nabla u|)=k_1u^p-k_2u^q$ and $u_0(\mathbf{x})\not \equiv 0$ a nonnegative function from $C^1({\bar{\Omega }})$, satisfying the compatibility condition $\frac{\partial u_0(\mathbf{x})}{\partial \varvec{\nu }}+h u_0(\mathbf{x})=0$ on $\partial \Omega$. Moreover, let $u\in C^{2,1}(\Omega \times (0,t^*))\cap C^{1,0}({\bar{\Omega }} \times [0,t^*))$ be the nonnegative solution of problem (2). If

$$\begin{aligned} \begin{aligned} \psi (t):&=-\frac{p+m}{2m}\int \limits _\Omega |\nabla u^m|^2 \mathrm{d} \mathbf{x}+ k_1 \int \limits _\Omega u^{p+m} \mathrm{d} \mathbf{x}\\&\quad -\,k_2\int \limits _\Omega u^{q+m} \mathrm{d} \mathbf{x}-\frac{h(p+m)}{2}\int \limits _{\partial \Omega } u^{2m}\mathrm{d}s, \quad \text { for all } t \in (0,t^*), \end{aligned} \end{aligned}$$

is such that $\psi (0)> 0$, then $t^*<\infty$, or equivalently $I=(0,t^*)$. In particular, $||u(\cdot ,t)||_{L^\infty (\Omega )}\nearrow \infty$ as $t\searrow t^*$ at some time $t^*$ satisfying, for $\varphi (t):=\int \limits _\Omega u^{m+1}d \mathbf{x}$,

$$\begin{aligned} t^* <T= \frac{1}{p-1}\frac{\varphi (0)}{\psi (0)}. \end{aligned}$$

Proof

Let u be the nonnegative classical solution of (2) satisfying $u_{ \varvec{\nu }}=-h u$ on $\partial \Omega$. By a differentiation, we can write

$$\begin{aligned} \begin{aligned} \psi '(t)&=-\frac{p+m}{m}\int \limits _\Omega \nabla u^m\cdot (\nabla u^m)_t \hbox {d}\mathbf{x}+k_1(p+m)\int \limits _\Omega u^{p+m-1}u_t \hbox {d} \mathbf{x}\\&\quad -\,k_2(q+m)\int \limits _\Omega u^{q+m-1}u_t \hbox {d} \mathbf{x}-h(p+m)m\int \limits _{\partial \Omega }u^{2m-1}u_t\hbox {d}s\\&=-\,\frac{p+m}{m}\int \limits _{\partial \Omega }(u^m)_t \nabla u^m \cdot \varvec{\nu }\hbox {d}s+ \frac{p+m}{m}\int \limits _\Omega (u^m)_t \Delta u^m \hbox {d} \mathbf{x}\\&\quad +\,k_1(p+m)\int \limits _\Omega u^{p+m-1}u_t \hbox {d} \mathbf{x}-k_2(q+m)\int \limits _\Omega u^{q+m-1}u_t \hbox {d} \mathbf{x}\\&\quad -\,h(p+m)m\int \limits _{\partial \Omega }u^{2m-1}u_t\hbox {d}s \\&\ge \,(p+m)\int \limits _\Omega u^{m-1}u_t(\Delta u^m+k_1u^p-k_2u^q) \hbox {d} \mathbf{x}\\&=(p+m)\int \limits _\Omega u^{m-1}(u_t)^2 \hbox {d} \mathbf{x}\ge 0\quad \text {for all }t\in (0,t^*), \end{aligned} \end{aligned}$$

(28)

where we have used the integration by parts formula and the assumption $p\ge q$.

Similarly, as to the evolution of $\varphi (t):=\int \limits _\Omega u^{m+1}d \mathbf{x}$, we derive

$$\begin{aligned} \begin{aligned} \frac{1}{m+1}\varphi '(t)&=\int \limits _\Omega u^m (\Delta u^m+k_1u^p-k_2u^q)\hbox {d}\mathbf{x}\\&=-\,\int \limits _\Omega \nabla u^m\cdot \nabla u^m \hbox {d} \mathbf{x}+k_1\int \limits _\Omega u^{p+m} \hbox {d} \mathbf{x}-k_2\int \limits _\Omega u^{q+m} \hbox {d} \mathbf{x}\\&\quad +\,\int \limits _{\partial \Omega }u^{m}\nabla u^m \cdot \varvec{\nu }\hbox {d}s\\ {}&=-\,\int \limits _\Omega |\nabla u^m|^2 \hbox {d} \mathbf{x}+k_1\int \limits _\Omega u^{p+m} \hbox {d} \mathbf{x}-k_2\int \limits _\Omega u^{q+m} \hbox {d} \mathbf{x}\\&\quad -\,mh\int \limits _{\partial \Omega }u^{2m}\hbox {d}s\\ {}&\ge \,-\frac{p+m}{2m}\int \limits _\Omega |\nabla u^m|^2 \hbox {d} \mathbf{x}+k_1\int \limits _\Omega u^{p+m} \hbox {d} \mathbf{x}-k_2\int \limits _\Omega u^{q+m} \hbox {d} \mathbf{x}\\&\quad -\, \frac{h(p+m)}{2}\int \limits _{\partial \Omega }u^{2m}\hbox {d}s=\psi (t) \quad \text { for all }t\in (0,t^*), \end{aligned} \end{aligned}$$

(29)

where in this case, we relied on the fact that $p\ge m$. Now, the hypotheses $\psi (0)> 0$, $\varphi (0)>0$, (28) and (29) yield

$$\begin{aligned} \psi '(t)>0, \;\psi (t)> 0,\; \varphi '(t)>0\; \text {and}\;\varphi (t)>0 \; \text {on} \quad (0,t^*). \end{aligned}$$

Since by the Young inequality, we have that for all $t\in (0,t^*)$

$$\begin{aligned} \frac{1}{m+1}\varphi '(t)=\int \limits _\Omega u^\frac{m+1}{2} u^\frac{m-1}{2}u_t\hbox {d} \mathbf{x}\le \bigg (\int \limits _\Omega u^{m+1}\hbox {d} \mathbf{x}\bigg )^\frac{1}{2} \bigg (\int \limits _\Omega u^{m-1}(u_t)^2 \hbox {d}\mathbf{x}\bigg )^\frac{1}{2}, \end{aligned}$$

this implies by virtue of the definition of $\varphi$, in conjunction with (28) and (29),

$$\begin{aligned} \varphi (t)\psi '(t)\ge \frac{m+p}{(m+1)^2}\varphi '(t)^2\ge \frac{m+p}{m+1}\psi (t)\varphi '(t) \quad \text {on} \quad (0,t^*), \end{aligned}$$

or equivalently

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}\big (\psi \varphi ^{-\frac{m+p}{m+1}}\big )\ge 0 \quad \text {on} \quad (0,t^*). \end{aligned}$$

Subsequently, an integration on (0, t) with $t<t^*$ infers

$$\begin{aligned} \psi (t)\ge \psi (0) \varphi (0)^{-\frac{m+p}{m+1}}\varphi (t)^{\frac{m+p}{m+1}}\quad \text {on} \quad (0,t). \end{aligned}$$

Finally, recalling (29), we have

$$\begin{aligned} \varphi '(t)\varphi (t)^{-\frac{m+p}{m+1}}\ge (m+1)\psi (0) \varphi (0)^{-\frac{m+p}{m+1}}\quad \text {on} \quad (0,t), \end{aligned}$$

and with $(m+p)/(m+1)>1$ a further integration leads to

$$\begin{aligned} \frac{1}{\varphi (t)^\frac{p-1}{m+1} }\le \frac{1}{\varphi (0)^\frac{p-1}{m+1}}-(p-1)\frac{\psi (0)}{\varphi (0)^{\frac{m+p}{m+1}}}t, \end{aligned}$$

that, by virtue of the positivity of $\varphi$, cannot hold for $t\ge T=\varphi (0)/((p-1)\psi (0))$. In conclusion, the extensibility criterion (3) implies that $I=(0,t^*)$, for some $t^*<T$. $\square$

Remark 2

As to the above technical requirement $\psi (0)>0$, we desire to emphasize its consistency with the thesis of Theorem 3.1. Indeed, for a given initial data $u_0$, with the suitable regularity and properties, such strictly positivity of $\psi (0)$ is essentially achieved when the value of the positive term $k_1\int \limits _\Omega u_0^{p+m}\hbox {d} \mathbf{x}$ dominates that from the other remaining contributions. Since this is certainly possible for $k_1$ sufficiently large, and $k_1$ is precisely the coefficient associated with the growth source in the model, the conclusion on the unboundedness of the solution u can be rather expected. Similar qualitative arguments apply imposing weak dampening and/or rate outflow effects, i.e., small values of $k_2$ and/or h. Finally, in support of the overall assumptions of Theorem 3.1, we mention [10, 11], where certain numerical methods capable to detect unbounded solutions to porous medium equations defined in bounded intervals of ${\mathbb {R}}$ and equipped with different boundary conditions are employed.

3.2 A criterion for global existence

In the next result, we are interested to examine the opposite situation described in Theorem 3.1. Precisely, by considering in system (2) again $g(u,|\nabla u|)=k_1u^p-k_2u^q$ and Robin boundary conditions, we establish that when the effect of the source (exponent p) is weaker than that of the diffusion (exponent m), the negative flux on the boundary prevents blow-up, even for arbitrary large initial data $u_0(\mathbf{x})$ and any small absorption effect (exponent q).

Theorem 3.2

Let $\Omega$ be a bounded smooth domain of ${\mathbb {R}}^N$, $N\ge 1,$ satisfying assumptions (${\mathcal {H}}_{1}$) and (${\mathcal {H}}_{2}$) . Moreover, let $k_1,k_2,h>0$, $k=1$, $q \ge 1$, $p<m$, with $m > 1$, $g(u,|\nabla u|)=k_1u^p-k_2u^q$ and $u_0(\mathbf{x})\not \equiv 0$ a nonnegative function from $C^0({\bar{\Omega }})$. Then the nonnegative solution $u\in C^{2,1}(\Omega \times (0,t^*))\cap C^{1,0}({\bar{\Omega }} \times [0,t^*))$ of problem (2) is global, or equivalently $I=(0,\infty )$.

Proof

Let u be the nonnegative classical solution of (2) satisfying $u_{ \varvec{\nu }}=-h u$ on $\partial \Omega$. By differentiating $\varphi (t):=\int \limits _\Omega u^{m+1} \hbox {d} \mathbf{x}$, we derive

$$\begin{aligned} \begin{aligned} \frac{1}{m+1}\varphi '(t)&=\int \limits _\Omega u^m (\Delta u^m+k_1u^p-k_2u^q)\hbox {d}\mathbf{x}\\&=-\,\int \limits _\Omega \nabla u^m\cdot \nabla u^m \hbox {d} \mathbf{x}+k_1\int \limits _\Omega u^{p+m} \hbox {d} \mathbf{x}-k_2\int \limits _\Omega u^{q+m} \hbox {d} \mathbf{x}\\&\quad +\,\int \limits _{\partial \Omega }u^{m}\nabla u^m \cdot \varvec{\nu }\hbox {d}s\\ {}&\le -\,\int \limits _\Omega |\nabla u^m|^2 \hbox {d} \mathbf{x}+k_1\int \limits _\Omega u^{p+m} \hbox {d} \mathbf{x}\quad \text {on}\;(0,t^*), \end{aligned} \end{aligned}$$

where we have neglected the last two nonpositive integrals.

On the other hand, since $p<m$, we have thanks to the Young inequality and for some $\varepsilon >0$

$$\begin{aligned} k_1\int \limits _\Omega u^{p+m} \hbox {d} \mathbf{x}\le \varepsilon \int \limits _\Omega u^{2m} \hbox {d} \mathbf{x}+ C(\varepsilon )|\Omega | \quad \text {on}\;(0,t^*), \end{aligned}$$

with $C(\varepsilon )=(2m \varepsilon /(p+m)k_1)^{(m+p)/(p-m)}(m-p)/2m>0$ (recall $m>p$). Subsequently,

$$\begin{aligned} \begin{aligned} \frac{1}{m+1}\varphi '(t)&\le -\int \limits _\Omega |\nabla u^m|^2 \hbox {d} \mathbf{x}+\varepsilon \int \limits _\Omega u^{2m} \hbox {d} \mathbf{x}+ C(\varepsilon )|\Omega | \\&\le \, -\sigma \int \limits _\Omega u^{2m} \hbox {d} \mathbf{x}+\varepsilon \int \limits _\Omega u^{2m} \hbox {d} \mathbf{x}+ C(\varepsilon )|\Omega |\\&=-\,(\sigma -\varepsilon ) \int \limits _\Omega u^{2m} \hbox {d} \mathbf{x}+ C(\varepsilon )|\Omega | \quad \text {on}\;(0,t^*), \end{aligned} \end{aligned}$$

where we have estimated the integral depending on $|\nabla u^m|^2$ by means of (11) of Lemma 2.2 with, of course, $V=u$. By choosing $\varepsilon =\sigma /2>0$, and by taking in consideration that an application of the Young inequality infers

$$\begin{aligned} -\int \limits _\Omega u^{2m} \hbox {d}\mathbf{x}\le - |\Omega |^\frac{1-m}{1+m} \varphi ^\frac{2m}{m+1}\quad \text {on}\;(0,t^*), \end{aligned}$$

the previous estimate reads

$$\begin{aligned} \varphi '(t)\le -C_0 \varphi ^\frac{2m}{m+1}(t)+C_1\quad \text {on}\;(0,t^*), \end{aligned}$$

where $C_0=(m+1)\sigma |\Omega |^\frac{1-m}{1+m}/2$ and $C_1=(m+1)C(\varepsilon )|\Omega |$; consequently, ODE comparison arguments justify that

$$\begin{aligned} \varphi (t)\le C:=\max \bigg \{\varphi (0),\bigg (\frac{C_1}{C_0}\bigg )^\frac{m+1}{2m}\bigg \}\quad \text {on}\;(0,t^*). \end{aligned}$$

Finally, well–known extension results for ODE’s with locally Lipschitz continuous right side (see, for instance, [18]), show that $t^*=\infty$; indeed, if $t^*$ were finite, $\varphi (t) \nearrow +\infty$ as $t\searrow t^*$ and it would contradict $\varphi (t)\le C$ on $(0,t^*).$ In conclusion, again the extensibility criterion (3) implies $I=(0,\infty )$. $\square$

Remark 3

Conversely to the demonstration of Theorem  3.1, evidently, the proof of this last theorem remains valid also for $k_2=0$, that is in the complete absence of absorption terms in g. In any case, we preferred to consider the expression of the function g in Theorem  3.2 as that in Theorem 3.1 exactly to better highlight the different behaviors of the corresponding solutions to problem (2) despite the same source.

In the behalf of scientific completeness, we mention [9, 24] where some questions concerning existence of local-in-time classical solutions to a class of systems tied to (2) are discussed under slightly more general boundary conditions but more specific sources than those considered in Theorems 3.1 and 3.2 ; hence, such theorems do not fall within these contributions.

3.3 Lower bounds of the blow-up time

This last theorem is concerned with lower bounds of the blow-up time $t^*$ for unbounded solutions to (2), when gradient nonlinearities with absorption effects appear in g. More precisely, we define $g(u,|\nabla u|)=k_1u^p-k_2|\nabla u|^q$ and endow the problem with Dirichlet boundary conditions. We are not aware of general results which straightforwardly infer the existence of unbounded solutions to system (2) under these hypotheses; nevertheless, in the spirit of the result derived in Theorem 3.1, for which blow-up occurs for large initial data and despite negative flux on the boundary, we understand that also in these circumstances seems reasonable to assume the existence of such blowing-up solutions.

Theorem 3.3

Let $\Omega$ be a bounded domain of ${\mathbb {R}}^3$ with Lipschitz boundary. Moreover, let $k_1,h>0$, $k=0$, $p,q,\alpha ,\beta ,\mu$ and $\gamma$ as in (13), $g(u,|\nabla u|)=k_1u^p-k_2|\nabla u|^q$ and $u_0(\mathbf{x})\not \equiv 0$ a nonnegative function from $C^0({\bar{\Omega }})$, satisfying the compatibility condition $u_0(\mathbf{x})=0$ on $\partial \Omega$. Hence, it is possible to find a positive number $\varSigma$ with the following property: If $k_2$ is a positive real satisfying

$$\begin{aligned} k_2\ge k_1(k_1\varSigma )^\frac{1-\mu }{\gamma -1}, \end{aligned}$$

(30)

and $u\in C^{2,1}(\Omega \times (0,t^*))\cap C^{0}({\bar{\Omega }} \times [0,t^*))$ is a nonnegative solution of (2) such that $W(t)\nearrow +\infty$ as $t\searrow t^{*}$, with some finite $t^{*}$ and

$$\begin{aligned} W(t) =\int \limits _{\Omega }u^{m(p-1)} \mathrm{d} \mathbf{x}, \end{aligned}$$

(31)

then

$$\begin{aligned} t^{*} \ge \frac{W(0)^{-\alpha \beta +1}}{({\mathcal {M}}W(0)^{(1-\beta )\alpha } +{\mathcal {N}})(-\alpha \beta +1)}, \end{aligned}$$

${\mathcal {M}}$ and ${\mathcal {N}}$ being two positive computable constants.

Proof

Let u be the nonnegative classical solution of (2) satisfying $u=0$ on $\partial \Omega$ and $t^*$ be the instant of time where the W-measure (31) associated with u becomes unbounded. For $s=p-1$, let us differentiate with respect to the time t such W-measure. Due to the divergence theorem and the boundary conditions, we obtain

$$\begin{aligned} \begin{aligned} W'(t)&= ms \int \limits _{\Omega } u^{ms-1}[\Delta (u^m)+k_1u^p -k_2|\nabla u|^q] \hbox {d}{} \mathbf{x} \\&=-\,ms \int \limits _{\Omega } \nabla u^{ms-1} \cdot \nabla (u^m) \hbox {d} x\\ {}&\;\;\;\;+\, msk_1 \int \limits _\Omega u^{s(m+1)} \hbox {d}{} \mathbf{x} -msk_2\int \limits _{\Omega } u^{ms-1}|\nabla u|^q \hbox {d}{} \mathbf{x}\\ {}&=-\,m^2 s(ms-1)\int \limits _\Omega u^{ms-3+m}|\nabla u|^2 \hbox {d}{} \mathbf{x} \\ {}&\;\;\;\;+\,msk_1 \int \limits _\Omega u^{s(m+1)} \hbox {d}{} \mathbf{x} -msk_2\int \limits _{\Omega } u^{ms-1}|\nabla u|^q \hbox {d}{} \mathbf{x} \quad \text {on} \quad (0,t^*). \end{aligned} \end{aligned}$$

(32)

Now, the statements given in (13) imply $ms+q-1>2$, so we can invoke inequality [29, (2.10)] achieving

$$\begin{aligned} \begin{aligned} msk_2\int \limits _{\Omega }u^{ms -1}|\nabla u|^q \hbox {d}{} \mathbf{x}&= msk_2\Big (\frac{q}{ms+q-1}\Big )^q\int \limits _\Omega |\nabla u^{\frac{ms+q-1}{q}}|^q \hbox {d}{} \mathbf{x} \\ {}&\ge \, msk_2\Big (\frac{2\sqrt{\lambda _1}}{ms + q-1}\Big )^q \int \limits _{\Omega }u^{ms + q-1} \hbox {d}{} \mathbf{x}\quad \text {on} \quad (0,t^*), \end{aligned} \end{aligned}$$

(33)

where $\lambda _1$ is the optimal Poincaré constant.

From now on, for simplicity, we indicate $u^s=V$ so to have

$$\begin{aligned} |\nabla V|^2= s^2 u^{2(s-1)}|\nabla u|^2. \end{aligned}$$

(34)

As a consequence, again due to the positions made in (13), it holds that $(m-2)+d>0$, so that using (33) and (34), relation (32) becomes

$$\begin{aligned} \begin{aligned} W'(t)&\le -c_1 \int \limits _{\Omega }V^{(m-2)+d}|\nabla V|^2 d\mathbf{x}+c_2 \int \limits _{\Omega } V^{m + 1}\hbox {d}{} \mathbf{x}\\&\quad -\,msk_2\Big (\frac{2\sqrt{\lambda _1}}{ms + q-1}\Big )^q \int \limits _{\Omega } V^{m + \mu }\hbox {d}{} \mathbf{x}\\ {}&=-\,c_3 \int \limits _{\Omega }|\nabla V^\frac{m+d}{2}|^2 \hbox {d}{} \mathbf{x}+c_2 \int \limits _{\Omega } V^{m + 1}\hbox {d}{} \mathbf{x} \\&\quad -\,msk_2\Big (\frac{2\sqrt{\lambda _1}}{ms + q-1}\Big )^q \int \limits _{\Omega } V^{m + \mu }\hbox {d}{} \mathbf{x}\quad \text {on} \quad (0,t^*), \end{aligned} \end{aligned}$$

(35)

where

$$\begin{aligned} c_1=\frac{m^2(ms-1)}{s},\quad c_2=msk_1,\quad c_3=\frac{4}{(m+d)^2}c_1. \end{aligned}$$

Now we are in the position to apply Lemma 2.3: by using relation (14) with $\varepsilon _1=k_2\big (2\sqrt{\lambda _1}/(ms + q-1)\big )^q(\gamma -\mu )/(k_1(\gamma -1))$, (35) is simplified to

$$\begin{aligned} W'(t)\le -c_3 \int \limits _{\Omega }|\nabla V^\frac{m+d}{2}|^2 \hbox {d}{} \mathbf{x}+c_4 \int \limits _{\Omega } V^{m + \gamma }\hbox {d}{} \mathbf{x}\quad \text {on} \quad (0,t^*), \end{aligned}$$

where

$$\begin{aligned} c_4=c_2\frac{1 - \mu }{\gamma - \mu } \varepsilon _1^{-\frac{\gamma - 1}{1 - \mu }}, \end{aligned}$$

while rearranging the term $\int \limits _{\Omega } V^{m + \gamma }\hbox {d}{} \mathbf{x}$ through (15) we obtain

$$\begin{aligned} \begin{aligned} W'(t)&\le \Big (c_5\varepsilon _2+c_6\varepsilon _2^{1-\frac{2(m+d)}{3\delta }}-c_3\Big ) \int \limits _\Omega |\nabla V^{\frac{m+d}{2}}|^2 \hbox {d}{} \mathbf{x}\\&\quad +\,{\mathcal {M}}\Big (\int \limits _\Omega V^m \hbox {d}\mathbf{x}\Big )^{\alpha }+{\mathcal {N}}\Big (\int \limits _\Omega V^m \hbox {d}\mathbf{x}\Big )^{\alpha \beta }\quad \text {on} \quad (0,t^*), \end{aligned} \end{aligned}$$

(36)

with

$$\begin{aligned} {\left\{ \begin{array}{ll} c_5=C_S^{\frac{3\delta }{m+d}+\frac{6\alpha }{2m+3d}}\frac{3\alpha dc_4\sigma }{2m+3d},&{} c_6= \frac{3\delta c_4 C_S^\frac{3\delta }{m+d}}{2(m+d)},\\ {\mathcal {M}}= C_S^\frac{3\delta }{m+d}(1-d)\varepsilon _2\sigma c_4,&{} {\mathcal {N}}= C_S^{\frac{3\delta }{m+d}+\frac{6\alpha }{2m+3d}} \frac{2m+3d-3\alpha }{2m+3d} \varepsilon _2 c_4d\sigma . \end{array}\right. } \end{aligned}$$

Hereafter, setting

$$\begin{aligned}\varSigma = \frac{\frac{C_S^{\frac{3\delta }{m+d}} s^2(m+d)^2}{4 m(ms-1)}}{ \Big [\frac{\gamma -\mu }{\gamma -1}\Big (\frac{2\sqrt{\lambda _1}}{ms+q-1}\Big )^q\Big ]^{\frac{\gamma -1}{1-\mu }}} \frac{1-\mu }{\gamma -\mu }\Big (\frac{6(m+d)C_S^{\frac{6\alpha }{2m+3d}}d \alpha \sigma }{(2m+3d)(2m+2d-3\delta )}\Big )^{1-\frac{3\delta }{2(m+d)}},\end{aligned}$$

we observe that using the values of the constants $c_1,c_2,\ldots ,c_6$ defined so far, relation (25) is precisely equivalent to (30). Subsequently, Lemma 2.4 warrants that for $\varepsilon _2=\xi _m$, whose value was computed in (27), $c_5\varepsilon _2+c_6\varepsilon _{2}^{1-2(m+d)/3\delta }-c_3\le 0$; for such a $\varepsilon _2$, and taking in mind (31), inequality (36) is simplified to

$$\begin{aligned} W'(t)\le {\mathcal {M}}\Big (\int \limits _\Omega V^m \hbox {d}\mathbf{x}\Big )^{\alpha }+{\mathcal {N}}\Big (\int \limits _\Omega V^m \hbox {d}\mathbf{x}\Big )^{\alpha \beta }={\mathcal {M}}W^\alpha +{\mathcal {N}}W^{\alpha \beta }\quad \text {on} \quad (0,t^*). \end{aligned}$$

(37)

Since we are assuming that $W(t)\nearrow \infty$ as $t\searrow t^{*}$, W(t) can be nondecreasing, so that $W(t) \ge W(0)>0$ with $t\in [0,t^{*})$, or nonincreasing (possibly presenting oscillations), so that there exists a time $t_1$ where $W(t_1)=W(0)$. In any case, we can write $W(t) \ge W(0)$ for all $t \in [t_1, t^{*})$, where $0\le t_1<t^{*}$. By virtue of (13), $\alpha ,\beta >1$, so that this implies that

$$\begin{aligned} W(t) \le W (0)^{1-\beta } W(t)^{\beta }, \quad t \in [t_1,t^{*}), \end{aligned}$$

which, in conjunction with (37), produces

$$\begin{aligned} W'(t) \le ({\mathcal {M}}W(0)^{(1-\beta )\alpha }+{\mathcal {N}})W^{\alpha \beta }, \quad t \,\in [t_1,t^{*}). \end{aligned}$$

(38)

Finally, integrating (38) between $t_1$ and $t^{*}$, we arrive at (recall $W(t_1)=W(0)$) the inequality

$$\begin{aligned} \begin{aligned} \frac{W(0)^{-\alpha \beta +1}}{-\alpha \beta +1}&=\frac{W(\tau )^{-\alpha \beta +1}}{-\alpha \beta +1}\bigg |^ {t^{*}}_{t_1} \le \int \limits _{t_1}^{t^{*}} ({\mathcal {M}}W(0)^{(1-\beta )\alpha }+{\mathcal {N}})\hbox {d}\tau \\&\le \, \int \limits _0^{t^{*}} ({\mathcal {M}}W(0)^{(1-\beta )\alpha }+{\mathcal {N}})\hbox {d}\tau = ({\mathcal {M}}W(0)^{(1-\beta )\alpha }+{\mathcal {N}})t^{*}, \end{aligned} \end{aligned}$$

which concludes the proof. $\square$

The previous theorem is a general extension of the result given in [40, Schaefer, 2008], where the gradient nonlinearity for g does not take part ($k_2=0$). Nevertheless, even though in the proof of Theorem 3.3 we used some ideas of [40], the presence of $|\nabla u|^q$ makes the demonstration more complex and requires other necessary derivations, which are somehow tricky and tedious. In particular, these further computations lead inter alia to consider the largeness assumption of $k_2$ (that is relation (30), not appearing in [40]).

4 Future works

In this concluding section, we leave open some questions, naturally arising from a proper examination of the paper.

• Validity of Theorem 3.3 when the Dirichlet boundary conditions are replaced by the Robin ones. Indeed, the proof of Theorem 3.3 counts on (15), which has been derived by virtue of the fact that $V=0$ on $\partial \Omega .$ If, conversely, $V_{ \varvec{\nu }}+h V=0$ on $\partial \Omega$ relation (16) is no longer valid and has to be modified adding to its right side a term involving $\int _\Omega V^{m+d} d\mathbf{x}$.

• As specified, in this research, the question of the existence of classical solutions to problem (2) has been circumvented, being these a priori given. Resorting to some weaker solutions concept (as expected for porous media equations), such existence could be possibly accomplished by appropriate regularizing actions on the original problem, the use of mollification techniques and general compactness arguments.

• The gradient nonlinearities $-|\nabla u|^q$ considered in our general nonlinear diffusion problem $(m>1)$ have an absorbing effect on the model. Conversely, in Sect. 1, we presented some papers dealing with the linear diffusion ($m=1$) Hamilton–Jacobi equation with first-order superquadratic Hamiltonian growth (i.e., $+|\nabla u|^q$, with $q>2$). Since the treatment of the porous medium equation is rather different and the extension to the range $m>1$ of general results obtained for $m=1$ is not an automatic process, the mentioned contributions could be of inspiration and support to see how far those analyses established in the framework of linear Hamilton–Jacobi equations with superquadratic growth can be generalized to nonlinear diffusion equations with the same growth.