Energy solutions to the bi-harmonic parabolic equations

1.   Introduction

This note investigates the initial value problem for the inhomogeneous non-linear fourth-order parabolic equation

The wave function is $u:(t, x)\in\mathbb R_+\times\mathbb R^N\to \mathbb{R}$ for some integer number $N\geq3$. The inhomogeneous nonlinear source term satisfies $p > 1$ and $\varrho > 0$.

The fourth-order parabolic problem models a variety of physical processes, such as phase transition, thin-film theory, and lubrication theory. In particular, it can be used to describe the evolution process of nanoscale thin films, with epitaxial growth; see, for instance, ^[7,10,13,20].

In recent years, fourth-order parabolic equations have been studied extensively. We refer the reader to the survey paper ^[2], where Section 14 includes some higher-order parabolic problems. The global well-posedness and finite-time blow-up properties of solutions have been investigated by many authors. See ^{[4,5,8,14,15,17,21]} and the references therein for the background for the study of bi-harmonic parabolic problems.

This note aims to obtain a threshold of global existence and exponential decay versus finite time blow-up of energy solutions to the inhomogeneous nonlinear bi-harmonic parabolic problem (IBNLH). The novelty is to consider the inhomogeneous regime $\varrho\neq0$, which complements the results in ^[19]. The method uses the standard stable sets under the flow of (IBNLH), due to Payne-Sattynger ^[12].

The plan of this note is as follows: Section 2 contains the main result and some standard estimates needed in the sequel. Section 3 proves the main result.

Let us recall the standard Lebesgue space

For $r\geq 1$, the usual Lebesgue norm reads

Finally, letting the standard Laplacian operator $\Delta: = \sum_{k = 1}^N\frac{\partial^2}{\partial x_k^2}$, we denote the following Sobolev space and its usual norm

2.   Background and main result

This section contains the main contribution of this note and some useful standard estimates.

2.1.   Preliminary

Let us denote the free bi-harmonic heat kernel

where $\mathcal{F}$ is the Fourrier transform. Thanks to the Duhamel formula, solutions to (IBNLH) satisfy the integral equation

If $u$ resolves the equation (IBNLH), then so does the family ${u}_\kappa: = \kappa^\frac{4-\varrho}{p-1}{u}(\kappa^{4}\cdot, \kappa \cdot), \kappa > 0$. Moreover, there is only one invariant Sobolev norm under the above dilatation, precisely

So, the heat problem (IBNLH) is said to be energy-sub-critical if

where, we take $p^c = \infty$ if $1\leq N\leq4$. Let us denote the so-called action and constraint

A solution to (IBNLH) formally satisfies

Let us denote the minimization problem

Then, it is known [18, Theorem 2.17] that $m > 0$ is reached in a so-called ground state

In the spirit of ^[12], one defines some stable sets under the flow of (IBNLH).

The so-called Strichartz estimates will be useful.

Definition 2.1. A couple of real numbers $(q, r)$ is said to be admissible if

Denote the set of admissible pairs by $\Lambda$. If $I$ is a time slab, one denotes the Strichartz spaces

The Strichartz estimates read as follows.

Proposition 2.1. Let $N \geq 1$ and $T > 0$. Then,

Proof. Let the free fourth order heat equation

Taking the Fourrier part of $u$, yields

It's known ^[1] that $\mathcal F^{-1}\Big(y\mapsto e^{-t|y|^{4}}\Big)(x) = \frac1{t^\frac N4}h(\frac x{t^{\frac14}})$ for a certain function $h$ satisfying $|h(y)|\lesssim e^{-d|y|^\frac43}$ for some $d > 0$. This implies that

By interpolation, yields $\|e^{-t\Delta^2}u_0\|_{L^r}\lesssim t^{-\frac N4(1-\frac2p)}\|u_0\|_{L^{r'}}$ for all $r\geq 2$. Thus, applying [6, Theorem 1.2], we get (2.12) and (2.13). Finally, (2.14) follows arguing as in [11, (3.19)]. □

Using a contraction argument via Proposition 2.1 and following lines in [3, Theorem 1.2], we obtain the existence of energy solutions to (IBNLH).

Proposition 2.2. Let $N\geq3$, $0 < \varrho < \min\{4, \frac N2\}$, $\max\{1, \frac{2(1-\varrho)}N\} < p < p^c$ and $u_0\in H^2$. Then, there exist $T: = T_{N, \varrho, p, \|u_0\|_{H^2}} > 0$, and a unique local solution of (IBNLH), in the space

We end this sub-section with a useful ordinary differential inequality result [9, Lemma 4.2].

Lemma 2.1. Letting a real decreasing function on $[0, \infty)$ such that

for certain $A > 0, B > 0$. Then, there exists $T > 0$ such that

From now on, we hide the time variable $t$ for simplicity, spreading it out only when necessary.

2.2.   Main result

The contribution of this note is the next threshold of global existence and exponential decay versus finite time blow-up of solutions to (IBNLH).

Theorem 2.1. Let $N\geq3$, $0 < \varrho <$ min $\{4, \frac N2\}$, max $\{1, \frac{2(1-\varrho)}N\} < p < p^c$ and $u_0\in H^2$. Take the maximal solution of (IBNLH), denoted by $u\in C([0, T^+), H^2)$.

1. If $u_0\in\mathcal{PS}^-$, then $T^+ < \infty$ and

2. If $u_0\in\mathcal{PS}^+$, then $T^+ = \infty$ and there is $\alpha > 0$ such that

In view of the results stated in the above theorem, some comments are in order.

● The existence of the energy solution to (IBNLH) is given by Proposition 2.2.

● The global solution with data in $\mathcal{PS}^+$ decays exponentially.

● Arguing as in [16, Lemma 5.1], it follows that $\mathcal{PS}^\pm$ are stable sets under the flow of (IBNLH).

● The above result complements ^[19] in the inhomogeneous regime, namely $\varrho\neq0$.

3.   Global/non global existence of energy solutions

In this section, we prove Theorem 2.1. Let us define, for $\lambda > 0, \tau > 0$, the real function on $t\in[0, T^+)$,

Taking account of (2.7), we compute the derivatives

Thus, by (2.4), (2.6), and (3.3), we obtain for $\lambda > (1+p)S(u_0)$,

So, (3.4) implies that

Let us denote the quantities

Compute for $X\in \mathbb{R}$, the polynomial

So, (3.9) implies that

Moreover, taking account of (3.4), we write

Take the real function

with a derivative

Moreover, by (3.11), we have

Integrating (3.14) in time after testing with $g'$, it follows that

Moreover,

So, (3.16) implies that

Thus, applying (2.16), we get $T^+ < \infty$ and $\lim_{t\to T^+}\int_0^t\|u(s)\|^2\, ds = \infty$. This proves the finite time blow-up (2.18). Now, if $u_0\in\mathcal{PS}^+$, then,

So, (3.18) implies that $\sup_{t\in[0, T^+)}\|u(t)\|_{H^2} < \sqrt{\frac{2m(1+p)}{p-1}}$ and $u$ is global. Thus, by the stability of $\mathcal{PS}^+$ under the flow of (IBNLH) we get

Let us define for $\gamma > 0$ some modified functional and sets as follows:

The next auxiliary result follows lines in [8, Preliminaries].

Lemma 3.1. The next properties hold.

1. $\lim_{\gamma\to0^+}m_\gamma = 0, \quad \lim_{\gamma\to+\infty}m_\gamma = -\infty$;

2. $\gamma\to m_\gamma$ is increasing on $[0, 1]$ and decreasing otherwise, and $m_1 = m$;

3. Let $u\in H^2$ satisfy $S(u) < m$ and $\gamma_1 < 1 < \gamma_2$ be roots of $m_\gamma = S(u)$; then, $K_\gamma(u)$ has a constant sign in $(\gamma_1, \gamma_2)$.

Now, by (2.7) via the last point in Lemma 3.1, we write for $\gamma\in(\gamma_1, 1)$,

Finally, (3.24) gives the requested estimate (2.19). This ends the proof of Theorem 2.1.

4.   Conclusions

This note gives a threshold of global existence and exponential decay versus finite time blow-up of energy solutions to the inhomogeneous nonlinear bi-harmonic parabolic problem (IBNLH). The novelty is to consider the inhomogeneous regime $\varrho\neq0$, which complements the results in ^[19]. The method uses the standard stable sets under the flow of (IBNLH), due to Payne-Sattynger ^[12].

Author contributions

Saleh Almuthaybiri: Formal analysis, funding acquisition; Tarek Saanouni: Project administration, resources, supervision, validation, review. All authors have read and approved the final version of the manuscript for publication.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The Researchers would like to thank the Deanship of Graduate Studies and Scientific Research at Qassim University for financial support (QU-APC-2024).

Conflict of interest

The authors declare no conflict of interest.