Blow-up in damped abstract nonlinear equations

1.   Introduction. An example of study.

We first consider the following generalized Boussinesq equation with linear damping and a nonlinear source term

on $\mathbb{R}^+\times\mathbb{R}^N$, $\mathbb{R}^+ \equiv [0,\infty)$, where $m^2 > 0, \delta \geq 0, \alpha_i > 0, \; i = 1,2,3,$ and $f$ is a nonlinear function of the unknown $u$.

In [2], Boussinesq obtained an approximate equation from the Euler equation to describe bidirectional solitary waves propagating on the free surface of a constant depth in irrotational motion. This a particular case without dissipation of the one-dimensional version of ${\bf (GB)}$ and it is called the classical Boussinesq equation, Since then, several modifications have been proposed. See [23,3] for historical remarks and the physical foundations of several generalizations of the original Boussinesq equation to model nonlinear wave propagation in elasticity and fluid mechanics.

For an evolution equation, like the Cauchy problem ${\bf (GB)}$, we have the following questions. With respect to a functional framework and in terms on the initial data, study:

● Local existence and uniqueness of solutions.

● Non-global existence: maximal time of existence $T_{MAX} < \infty$.

● Global existence: $T_{MAX} = \infty$.

● In the latter case, the behavior of the solution as time approaches infinity.

Here, we shall study the second point. In fact, we will give conditions on the initial data, $u_0,u_1$, for which the corresponding solution, in some sense given later, exists only up to a finite time. In order to do that, we are going to consider an abstract formulation of problem ${\bf (GB)}$, so that can be applied to other equations.

2.   The abstract formulation, related results and preliminaries

We consider the following abstract differential equation.

For every initial data $u_0,u_1$, find $t \mapsto u(t), \; t \geq 0 \;$, such that

Here, $u_t \equiv \frac{d}{dt} u$, and we assume that the operators

are linear, continuous, positive and symmetric, and

are linear subspaces of the Hilbert space $H$ with inner product $(\cdot,\cdot)$ and norm $\|\cdot\|$. Here, $H^{'}, W_{P}^{'}, V_{A}^{'},$ are the corresponding dual spaces and we identify $H = H^{'}$. Then,

By means of the operators $P$ and $A$, we define the following bilinear forms

and corresponding norms

The following problem ${\bf (GB)^*}$ has the abstract form ${\bf (P)}$. Indeed, we remember that $(- \Delta)^{-s}u = \mathcal{F}^{-1}(|\omega|^{-2s}(\mathcal{F}u)(\omega))$, for $s > 0$, where $\mathcal{F}, \mathcal{F}^{-1}$ are the Fourier transform and the inverse Fourier transform, respectively. Applying $(- \Delta)^{-1}$, to ${\bf (GB)}$, we get

on $\mathbb{R}^+\times\mathbb{R}^N$.

Then, as usual $u(t) \equiv u(t,\cdot)$, and we define $H = L_2(\mathbb{R}^N)$, with inner product $(\cdot,\cdot)_2$ and norm $\|\cdot\|_2$. Also,

are defined, respectively, on the subspaces of $H$

Moreover, if

then

The analysis of problem ${\bf (P)}$ is in the framework of weak solutions in the sense of the following Definition 2.1. To this end, we consider the phase space

with norm

and we assume that the following hypotheses hold.

There exists $c > 0$, such that

The nonlinear source term $f: V_{A} \rightarrow H$, is such that $f(0) = 0$, and it is a potential operator with potential $F: V_{A} \rightarrow \mathbb{R}$, that is, $f(u) = D_u F(u)$. Moreover, they satisfy

and some constant $r > 2 + \delta/\sqrt{c}$.

Definition 2.1. For every initial data $(u_0,u_1) \in \mathcal{H}$, the map $(u_0,u_1)) \mapsto (u(t),\dot{u}(t)) \in \mathcal{H},$ where $\dot{u}(t) \equiv \frac{d}{dt} u(t)$, is a weak local solution of problem ${\bf (P)}$, if there exists some $T > 0$, such that $(u,\dot{u}) \in C([0,T);\mathcal{H})$, with $u(0) = u_0, \; \dot{u}(0) = u_1$, and

a. e. in $(0,T)$ and for every $w \in V_{A}$. We shall consider that the solution in this sense is unique and satisfies the following energy equation for $T > t \geq t_0 \geq 0$,

Furthermore, if the maximal time of existence $T_{MAX} < \infty$ then

equivalently, by the energy equation,

For the Boussinesq equation, hypothesis $(H0)$ holds with $c = \min\{m^2,\frac{\alpha_1}{\alpha_2}\}$. We assume that the source terms $f$ and the corresponding potential operator $F$ do not have any particular form but they satisfy $(H1)$. The existence and uniqueness of weak solutions of ${\bf (GB)^*}$, in the sense of Definition 2.1 holds, see [29,25,33].

An important set of solutions are the equilibria, that is solutions independent of time, that is, $\dot{u} = 0$. In this case, $u$ satisfies

for every $w \in V_{A}$. In particular, for $w = u$,

and then

By $(H0)$, $(u,0) \equiv (0,0)$ is an equilibrium. The set of nonzero equilibria, denoted by $\mathcal{E}$, with minimal energy is called ground state, and the corresponding value of the energy is the mountain pass level denoted by $d$, see [32]. Indeed,

where

is the Nehari manifold. By means of this number, the potential well method has been used to characterize the qualitative behavior for solutions of some equations of the type ${\bf (P)}$. For dissipative equations, $\delta > 0$, the behavior is completely different from the conservative case $\delta = 0$. Indeed, consider the initial and homogeneous Dirichlet boundary value problem of the nonlinear wave equation, which is a particular case of problem ${\bf (P)}$.

where $\Omega \subset \mathbb{R}^N$ is a bounded domain with smooth boundary $\partial \Omega$, $t \in \mathbb{R}^+$, and $\alpha > 0, \; g(u_t) = \delta u_t, \; \delta > 0, \; f(u) = \mu |u|^{r-2}u, \; r > 2$. Here,

Hypothesis $(H0)$ holds because of the Sobolev-Poincaré inequality,

and $(H1)$ holds if $r > 2 + \delta/\sqrt{c}$, with $rF(u) = \mu \|u\|_r^r$, where $\|\cdot\|_r$ is the norm in $L_r(\Omega)$. The following result about the dynamics of ${\bf (NLW)}$ holds.

Theorem 2.2. ([7,14]). For every solution $(u,\dot{u})$ of ${\bf (NLW)}$, in the sense of Definition 2.1, with initial data $(u_0,u_1) \in \mathcal{H} = V_{A} \times W_{P} \equiv H_0^1(\Omega) \times L_2(\Omega)$, only one of the following holds.

● There exists some $t_0 \geq 0$ such that $E(t_0) < d$, then either

and the solution blows up in a finite time or

and the solution is global, uniformly bounded in time in the norm of $\mathcal{H}$, and

exponentially in time.

● $E(t) \geq d$, for all $t \geq 0$, then the solution is global, uniformly bounded in time in the norm of $\mathcal{H}$, and

strongly in $\mathcal{H}$, where

We notice that the only way to get blow up of in finite time is that, along the solution and since the energy is a non-increasing function, there exists some $t_0 \geq 0$ such that $E(t_0) < d$ and $u(t_0) \in V$. If the damping term is nonlinear of the form $g(u_t) \equiv \delta |u_t|^{\lambda}u_t, \; \lambda > 2$, although the dynamics is more complicated, a similar result holds, see [8]. Indeed, the solution blows up in a a finite time if and only if $r > \lambda$ and there exists some $t_0 \geq 0$ such that $E(t_0) < d$ and $u(t_0) \in V$. An open problem is to characterize the set of initial data such that $E(t_0) < d$ and $u(t_0) \in V$, for some $t_0 > 0$, especially when $E_0 \equiv E(0) \geq d$ and $u_0 \notin V$. Another open problem is to find conditions on the initial data in order to get $E(t) \geq d$, for all $t \geq 0$. Then, according with Theorem 2.2 those solutions are global and they approach to the set of nonzero equilibria. See also [9], where Theorem 2.2 is proved for the Timoshenko equation. In the conservative case, $\delta = 0$, we know the following result for $E_0 \leq d$.

Theorem 2.3 ([24,10]). Consider any solution $(u,\dot{u})$ of ${\bf (NLW)}$, in the sense of Definition 2.1, with initial data $(u_0,u_1) \in \mathcal{H} = V_{A} \times W_{P} \equiv H_0^1(\Omega) \times L_2(\Omega)$ and $\delta = 0.$

If $E_0 < d$, then

● the solution blows up in a finite time if and only if $u_0 \in V$.

● the solution is global if and only if $u_0 \in W$, and in this case the solution is uniformly bounded in time in the norm of $\mathcal{H}$.

If $E_0 = d$, and

● there exists some $t_0 \geq 0$ such that $I(u(t_0)) > 0$, then the solution is global and uniformly bounded in time in the norm of $\mathcal{H}$.

● there exists some $t_0 \geq 0$ such that $I(u(t_0)) < 0$ and $(u(t_0), v(t_0))_2 \geq 0$, then the solution blows up in finite time.

The asymptotic behavior of the global solutions with $E_0 \leq d$ is unknown, as well as the characterization of the qualitative behavior for $E_0 > d$ in terms of the initial data. We have to mention two remarkable papers, [18,27] where the blow up of solutions for the Cauchy problem associated to ${\bf (NLW)}$ is studied, that is with $\Omega = \mathbb{R}^N$. In those works a nonlinear damping term of the form $g(u_t) \equiv \delta |u_t|^{\lambda}u_t$ is considered and it is proved that for any $(\eta,\xi) \in (0,\infty) \times [0,\infty)$, there are infinitely many compactly supported initial data $(u_0,u_1)$ such that if $\|\nabla u_0\|_2 = \eta$ and $E_0 = \xi$, the corresponding solution blows up in finite time. We notice that this result is not true for the initial boundary value problem ${\bf (NLW)}$ with that nonlinearity $g(u_t)$. Indeed, for $\xi < d$ and $\eta^2 < \frac{2rd}{\alpha(r - 2)}$ the solution is globally defined, see [8], because $u_0 \in W$ if and only if $\alpha \|\nabla u_0\|_2^2 < \frac{2r}{r - 2}d$. However, the result in [18,27] exhibits a complex dynamics because it is known that there are global solutions for the Cauchy problem associated to ${\bf (NLW)}$ with initial data such that $\eta$ and $\xi$ are small, see [27] and references therein.

For the ${\bf (GB)^*}$ problem, characterizations of blow-up and global solutions with $E_0 < d$ by means of the potential well method were proved in [21,22,16] and in [34], for the one-dimensional and the multidimensional problems, respectively. For $E_0 \geq d$, there are only partial results on the dynamics of particular equations of the type ${\bf (GB)^*}$. Most of them are results for blow up of solutions, see [29,25,16]. Similar analysis, for $E_0 \geq d$, have been performed to prove the same qualitative properties for other equations, see [14,28] to cite just some of the most influential papers on the subject, and see [4,19,20,26,30,31,35,36,37] and references therein for some recent works. In [6] a numerical study for the Cauchy problem of the focusing cubic nonlinear Klein-Gordon equation in three dimensions for radial initial data shows that for high energies, the qualitative behavior seems to be much more complicated than for $E < d$, and more research is required to find a threshold between globality and blow up of solutions. In [17] some numerical experiments were performed for the one-dimensional case of ${\bf (GB)^*}$, with $\delta = 0$, to calculate the mountain pass level and confirm the range of validity of the potential well method. In particular, for $E_0 > d$, it was observed that the sign of $I(u_0)$ is irrelevant to get the blow up of solutions in finite time, but instead, the sign of the inner product $\mathcal{P}(u_0,u_1)$ is important to obtain nonexistence of global solution. In [11,12] we have proved that this is true for the undamped nonlinear Klein-Gordon equation and for the problem ${\bf (P)}$ with $\delta = 0$. Here, we are going to prove that, for any $\delta > 0$ and any positive $E_0$, the sign of $\mathcal{P}(u_0,u_1)$ is the main ingredient to prove the nonexistence of global solutions for problem ${\bf (P)}$ and in particular for ${\bf (GB)^*}$. However, we shall consider $\delta \geq 0$, in order to see the influence of the damping in the qualitative behavior.

3.   Nonexistence of global solutions

We define

where $c > 0$ is the constant in $(H0)$, $\delta \geq 0$ is the damping coefficient, we have assumed that $r > 2 + \delta/\sqrt{c}$, see $(H1)$, and from the orthogonal decomposition

we have that

We also define the following functions

If $\mathcal{P}(u,\dot{u}) > 0$ and for fixed $\delta$, we notice that $q \mapsto \eta_{q,\delta}$ is strictly increasing, $\lambda \mapsto \mu_{\lambda,\delta}$ is strictly decreasing and $\nu \mapsto \sigma_{\nu}$ is strictly decreasing. They have the following relations

and, $\sigma_{\nu}(u,\dot{u}) < \eta_{q,\delta}(u,\dot{u})|_{q = 0} < \eta_{q,\delta}(u,\dot{u})|_{q = \frac{r - 2 - \delta/\sqrt{c}}{2}} < \mu_{\lambda,\delta}(u,\dot{u})$.

Now, we are able to present the main result of this work.

Theorem 3.1. Consider any solution of problem (P) in the sense of Definition 2.1. Assume that hypotheses $(H0)$ and $(H1)$ hold, and that

are satisfied. Then, there exists a nonempty interval

where, by means of the functions $\sigma_{\nu,\delta}(u_0,u_1)$ and $\mu_{\lambda,\delta}(u_0,u_1)$, we can obtain the size of such interval

for some $\frac{r - 2 - \delta/\sqrt{c}}{r - \delta/\sqrt{c}} < \lambda^{*} < 1$ and $\nu^{*} > 1$, with the following consequences:

(i) If the initial energy is such that $E_0 \in \mathcal{I}_\delta$, then the maximal time of existence of the solution is finite.

(ii) For fixed $\Psi(u_0)$ and $\delta$,

is strictly increasing, and we have the limit values

Corollary 1. Assume that hypotheses of Theorem 3.1 are met. For every number $\mathcal{K} > 0$, we can choose initial data with $\mathcal{P}(u_0,u_1)$ large enough, so that $\mathcal{K} \in \mathcal{I}_\delta$, and then the corresponding solution with $E_0 = \mathcal{K}$ exists only up to a finite time.

4.   Proofs

Proof. (of Theorem 3.1.) We assume that $\Psi(u(t)) \equiv \|u\|_{W_P}^2$ exists for any $t \geq 0$. If such a thing does not happen, then the solution cannot be global. We observe that

and then we get the following estimate

Hence,

By energy equation and hypotheses $(H0)$ and $(H1)$, we get the following chain of estimates

Hence, by (3),

We define

Consequently, from (4),

That is,

Due to (2) and since $r > 2 + \delta/\sqrt{c}$, see $(H1)$, we have that the following inequality is true

for any $t \geq 0$ sufficiently close to zero. That is,

Now, we multiply (5) by $\frac{d}{dt} \mathcal{F}(t) < 0$. Then, after an easy integration, we get

where

We shall prove that there exists a constant $\kappa_0 > 0$ such that

and then

Hence,

Which is impossible for any $t > \frac{\mathcal{F}(0)}{\kappa_0}.$ Then, the solution only exits up to a finite time.

Next, we prove that (7) holds. To this end, we consider the right hand side of (6) and define, for $s \geq 0$,

and we notice that $\mathcal{G}$ attains an absolute minimum, that is

with $s_0 \equiv \left(\frac{c(r - 2) - \delta \sqrt{c}}{2r E_0}\right)^{\left(\frac{r - 2 - \delta/\sqrt{c}}{2}\right)} > 0$, since $c(r - 2) - \delta \sqrt{c} > 0$, and

On the other hand,

Notice that (7) is satisfied for $\kappa_0^2 = \mathcal{G}(s_0)$, and $\mathcal{G}(s_0) > 0$ holds if and only if

which is equivalent to

where, we remember that

We consider the left hand side of (8) to define, for $s \geq 0$,

We observe that (8) is satisfied if and only if

We notice that $\mathcal{J}(s) \rightarrow \infty$ as, either $s \rightarrow 0$ or $s \rightarrow \infty$, and $\mathcal{J}$ attains an absolute minimum, that is

for $s_1 \equiv \left(\frac{c(r - 2) - \delta \sqrt{c}}{2r}\right) \Psi(u_0)$. Moreover, by (2), there exist exactly two different roots of $\mathcal{J}(s) = \frac{1}{2} \Phi(u_0,u_1)$, denoted by $\alpha_\delta$ and $\beta_\delta$, such that

and

And since $\mathcal{J}$ is strictly monotone for $s < s_1$ and $s > s_1$, it follows that, for fixed $\Psi(u_0)$, the interval $\mathcal{I}_\delta$ grows as $\mathcal{P}(u_0,u_1)$ grows. Precisely,

Then, (8) holds if and only if the initial energy satisfies (9). That is, if and only if $E_0 \in \mathcal{I}_\delta$. This proves that the maximum time of existence must be finite if the initial energy is within this interval.

We shall use the functions $\sigma_\nu$ and $\mu_{\lambda,\delta}$ to find the values of $\alpha_\delta$ and $\beta_\delta$, respectively. Remember that these are the roots of $\mathcal{J}(s) = \frac{1}{2} \Phi(u_0,u_1)$. To find $\alpha_\delta$, we consider the equation

where

is defined for $\nu > 1$. Notice that (10) holds if and only if

Which is equivalent to

We consider the function, defined for $s > 0$,

and notice that

as $\nu \rightarrow 0$ and $\nu \rightarrow \infty$. Furthermore, $f$ has an absolute minimum, that is

Moreover, from (2) and the definition of $\Phi(u_0,u_1)$,

Then, equation (11) equivalently (10), has two roots and only one bigger than one, that is, $\nu^{*} > 1$. Furthermore, at this root,

and

Next, we consider the equation

where

is defined for $0 < \lambda < 1$. We observe that (12) holds if and only if

And this is characterized by

Which is equivalent to

We consider the functions, defined for $s \in [s_2,1]$, with $s_2 = \frac{r - 2 - \delta/\sqrt{c}}{r - \delta/\sqrt{c}}$,

and

are strictly monotone increasing, and

since, from (2) and definition of $\Phi(u_0,u_1)$,

Then, there exists one and only one $\lambda^{*} \in \left(\frac{r - 2 - \delta/\sqrt{c}}{r - \delta/\sqrt{c}},1\right)$ where $g(\lambda^{*}) = h(\lambda^{*})$. That is, only one root $\lambda^{*}$ of equation (13), equivalently (12). Moreover,

and

Proof. (of Corollary 1.) Since

then, for every $\mathcal{K} > 0$ there exists $\mathcal{L} > 0$, such that

Hence, the corresponding solution with initial energy $E_0 = \mathcal{K}$ satisfying (2) exists only up to a finite time.

5.   Discussion and some remarks

Remark 1. It is well known that when the potential well method is applied, for $E_0 < d$, the qualitative behavior is determinated by the sign of $I(u_0)$. In particular, the blow-up of a solution is characterized if the initial data are such that $I(u_0) < 0$. For any positive value of the initial energy, this is not the case. Indeed, under the assumptions of Theorem 3.1, we get from energy equation, if $E_0 < \beta_\delta$, and since $\frac{r - 2 - \delta/\sqrt{c}}{r - \delta/\sqrt{c}} < \lambda^{*}_\delta < 1$, the following consequences

Hence,

Let us assume that the source term is large enough, that is,

then, by (14),

From $(H1)$, (15) holds if the source term is such that

Then, in this case the inequality $I(u_0) < 0$ is a necessary condition for nonexistence of global solutions. However, it seems that the condition $I(u_0) < 0$, alone, does not imply nonexistence of global solutions for high energies, see [14,28]. Moreover, the sign of $I(u_0)$ is not required in the proof of Theorem 3.1.

By $(H1)$ and the energy equation

If $E_0 < d$, the mountain pass level defined in (1), and $f(u) \equiv |u|^{r - 2}u$, then $rF(u) = \|u\|^r_r$, and

From Corollary 1, global nonexistence is obtained if $\mathcal{P}(u_0,u_1) > 0$. Hence, in order to conclude global nonexistence for $E_0 < d$, it is necessary that

We remember that for $E_0 < d$, see [7], we have that $I(u_0) < 0$ if and only if $2rd < (r - 2)\|u_0\|^r_r$. Apparently, only for energies $E_0 < d$, the condition $I(u_0) < 0$ characterizes the nonexistence of global solutions of problem ${\bf (P)}$.

Remark 2. We shall prove the following lower bound for $\beta_\delta$

which is equivalent to

To this end, consider first the following inequality

Notice that it is equivalent to

where $\frac{r - 2 - c/\sqrt{c}}{r - c/\sqrt{c}} < \lambda^* < 1$.

In order to prove last inequality, let us define for any $s > 0$, the positive function

and, from the proof of Theorem 3.1, we remember that $\lambda^{*}$ is a function of $s \equiv |\mathcal{P}(u_0,u_1)|^2$, defined implicitly by

where

Also, from the proof of Theorem 3.1 we know that

Then, from the definition of $\lambda^{*}$ and $\Phi_0$,

Also,

Consequently, there is some $s^* \in (0,\infty)$, such that $l(s^*) = \max_{s \in (0,\infty)} l(s)$. After some calculations, we find that

and consequently

Hence,

where

Notice that (16) is equivalent to

where

since $\lambda^* < 1.$

Consequently,

Remark 3. We shall prove the following upper bound for $\alpha_\delta$

To this end, notice that this inequality is equivalent to

where $\nu^{*} > 1$.

In order to prove last inequality, we define for any $s > 0$, the function

where we remember from the proof of Theorem 3.1 that $\nu^{*}$ depends of $s \equiv |\mathcal{P}(u_0,u_1)|^2$, defined implicitly by

where like in Remark 2

Furthermore, from the proof of Theorem 3.1 we know that

Now, from the definition of $\nu^{*}$ and $\Phi_0$,

Also,

After some calculations, it follows that,

Moreover,

Then, $l(s)$ is strictly increasing and bounded by

And this is equivalent to

Remark 4. The length of $\mathcal{I}_\delta$ depends on the size of the damping coefficient $\delta \geq 0$ and decreases as $\delta$ grows. Therefore, as the damping coefficient increases, the set of initial energies where we can have global non existence becomes smaller. In fact, from Remarks 2 and 3, we have that

In [1], the abstract problem ${\bf(P)}$ is studied with a more general linear damping term. In that paper, the nonexistence of global solutions is analyzed for any positive value of the initial energy. In fact, it is proved that the following functional, written in terms of our problem ${\bf(P)}$,

blows up in finite time if $(H0), (H1)$ and

hold. We notice that the last condition implies $\|u_0\|_{W_P} > 0, \; \mathcal{P}(u_0,u_1) > 0$, and that for large values of $E_0$, necessarily correspond large values of $\mathcal{P}(u_0,u_1)$, conclusion that agrees with our results.

6.   Examples

6.1.   Generalized Boussinesq equation

We consider the equation ${\bf (GB)^*}$. We remember that hypothesis $(H0)$ holds with $c = \min\{m,\frac{\alpha_1}{\alpha_2}\}$. Also, we assumed that the source terms $f$ and the corresponding potential operator $F$ do not have any particular form but they satisfy $(H1)$. The solution in the sense of Definition 2.1 holds, see [29,25,33], then nonexistence of global solutions is due to blow-up. By Theorem 3.1, if the initial data are such that

and the initial energy

is such that $E_0 \in \mathcal{I}_\delta = (\alpha_\delta,\beta_\delta)$, given in Theorem 3.1, then the corresponding solution is not global and blows up in finite time. Moreover, by Corollary 1, for every positive initial energy $E_0$, there exists initial data such that imply the nonexistence of global solutions in the norm of $\mathcal{H}$.

There are several results in the literature showing blow up for large positive values of the initial energy for equations of the type ${\bf (GB)^*}$. They consider either $\delta = 0$, see [16,4,36], or a linear damping term, which can be either weak or strong, see [25,26,29,30,31]. In most of them, the blow up is proved under the assumption (2) and if the initial energy satisfies an inequality of the type

with $C_j \geq 0, \; j = 1,2, \;\; C_1 + C_2 > 0$, are numbers depending on the damping constant $\delta$ and on the numbers $c,r$ in $(H0), (H1)$. See Remark 2 to compare this condition with a lower bound of $\beta_\delta.$

In [16], the one dimensional equation ${\bf (GB)^*}$ with a source term of polynomial type, $\delta = 0$ and $m^2 = 1 = \alpha_j, \; j = 1,2,3$, and consequently with $c = 1$, is considered. There, blow up is proved if $E_0$ satisfies, implicitly, the condition given in (8).

Very recently, characterizations for blow up and globality where given in [5] for a one dimensional sixth order nonlinear double dispersive equation with a linear restoring force and with $\delta = 0$. This equation is of the type ${\bf (P)}$, with

defined on the subspace of $H = L_2(\mathbb{R}^N)$,

with norms

where

Hence $c = 1$ in $(H0)$ and the source term is of polynomial type and $(H1)$ is satisfied. The characterizations are as follows.

Theorem 6.1 ([5]). For this particular example, the following characterizations hold.

● The solution blows up in a finite time, $T_{MAX} < \infty$, if and only if

● The solution is global, $T_{MAX} = \infty$, if and only if

Those conditions are not easy to verify. Hence, sufficient conditions in terms of the initial data are needed. In [5], blow up in finite time is guaranteed if

We observe that this inequality is not of the type (17), like the upper bound of $\beta_\delta$ in Remark 2. However for $\mathcal{P}(u_0,u_1)$ large enough, $\beta_\delta$ is larger than (18).

6.2.   Nonlinear Klein-Gordon and wave equations

Now, we consider the following problem with a linear damping term

on $\mathbb{R}^{+}\times\mathbb{R}^N$, where $\delta \geq 0, \; m^2 > 0.$

This equation was studied in [28] with $\delta = 0$. Here,

Hypothesis $(H0)$ holds with $c = \min\{1,m^2\}$, and $(H1)$ holds since $f$ is assumed of a polynomial type.

The solution in the sense of Definition 2.1 holds and nonexistence of global solutions is due to blow-up, see [28] for the details in the undamped case. Consequently, by Theorem 3.1, if the initial data satisfy

and the initial energy is such that $E_0 \in \mathcal{I}_\delta$, the corresponding solution blows up in finite time in the norm of $\mathcal{H}$. And by Corollary 1 for every positive initial energy $E_0$, there exist initial data satisfying last inequality such that the solution blows up.

In [28], blow up was proved under the assumption (2), and if the following inequalities hold

We observe that in Theorem 3.1 we did not assume any sign of $I(u_0)$ and $\beta_\delta$ is larger than the upper bound on $E_0$ given in [28], see Remark 2.

The one dimensional case of ${\bf (KG)}$, with $\delta = 0$ and $m^2 = 1$, was studied in [4] and blow up is showed under the assumption (2), and if the initial energy satisfies the implicit inequality given in (8).

In [37] the following initial and homogeneous Dirichlet boundary value problem of a nonlinear wave equation of the type ${\bf (KG)}$, with a strong linear damping term, a source $f$ of a polynomial type and $m^2 = 0$, is studied.

where $\Omega \subset \mathbb{R}^N$, is a bounded domain with smooth boundary

The blow up is proved under the assumption (2), and if the initial energy satisfies the inequality

where $C > 0$, depends on $\omega, \delta$ and $c,r$ in $(H0), (H1)$, and in the case that $\omega = 0$, the constant is equal to $C = \frac{c(r - 2)}{r(1 + \delta)}$. This inequality is not of the type of the upper bound of $\beta_\delta$ in Remark 2, but for $(u_0,u_1)_2$ large enough, $\beta_\delta$ is larger than the bound in (19).

6.3.   Kirchhoff equation

Many works have been published about the Kirchhoff equation. In particular, in [15] blow up of solutions of the following problem was studied.

where $\Omega \subset \mathbb{R}^N$, is a bounded domain with smooth boundary

In [15], existence and uniqueness is proved in the phase space

for a set of functions $g$ and $\phi$, satisfying several conditions. In particular, $g$ satisfies $(H1)$, that is,

where $G$ is the potential of $g$, and a typical form of $\phi$ is

with $c_j > 0, \; j = 1,2$ and $q \geq 1$. In this case, if $c_1 = c_2 = 1$ for simplicity,

has the potential

and satisfies $(H1)$ with

that is, the nonlinearity of the source term $g$ is stronger than the one of $\phi$. In contrast, for a Timoshenko equation where these two nonlinearities appear, if $\theta < 2(q + 1)$ holds then all the solutions are global and uniformly bounded in the phase space $\mathcal{H}$ defined above. Furthermore, all the solutions are attracted by the set of equilibria as times goes to infinity, see [13].

The blow up result showed in [15] holds if the initial data are such that

Note that this upper bound of $E_0$ is smaller than

which is of the type (17). And also, it is smaller than $\beta_\delta$ in Theorem 3.1.

Acknowledgments

I thank Professor Howard Levine for calling our attention to his work with Professor Todorova, [18]. We also thank to Professor Varga Kalantarov for sharing his articles, especially [1]. Finally, we thank to the anonymous referees for their valuable comments that improved the final form of this article. This work was supported by the Universidad Autónoma Metropolitana, Unidad Azcapotzalco.