Strong solutions and trajectory attractors to the thin-film equation with absorption

Abstract

We prove local and global in time existence of non-negative weak solutions to the thin-film equation with absorption and obtain sufficient conditions for extra regularity of these solutions. Moreover, for the class of global strong solutions, we show existence of a trajectory attractor.

Introduction

Thin liquid film models originate from numerous industrial applications connected with liquid dynamics, material science, medicine and biology (see [22]). Let $-\infty <a<b<\infty$, $n⩾1$, $\lambda ,\beta >0$ be arbitrary fixed and $u_0,h\in H^1(a,b)$. To simplify our notations, we will sometimes use $H^1$ for $H^1(a,b)$ and $H^2$ for $H^2(a,b)$. Assume that

$$u_0(x)⩾0\text{ and }h(x)⩾0\text{ for each }x\in (a,b).$$

Moreover, let

$$g(z):=|z{|}^{\lambda -1}z+\beta z,l(x,z):=g(z)-h(x)\forall (x,z)\in (a,b)\times {\mathbb{R}}^1.$$

In this paper, we analyse the existence, regularity and asymptotic behaviour of solutions to the following initial boundary value problem:

$$u_t+{(|u{|}^n{u}_{xxx})}_x+l(x,u)=0\text{ in }{Q}_T:=(a,b)\times (0,T),$$

$$u_x{|}_{x=a,b}=u_{xxx}{|}_{x=a,b}=0\text{ on }(0,T),$$

$$u{|}_{t=0}=u_0\text{ on }(a,b).$$

Here, the term $l(x,u)$ models a concurrence between non-linear absorption and spatial injection. The absorption-type term was first introduced in [15]. Dependence of solutions on the absorption term was studied in [26], [27], [28] and [11], [12]. The injection term first appeared in [23] and its influence on solution properties was not studied enough. Equation (1.3) is a nonlinear degenerate fourth-order parabolic PDE and it models a thin liquid film on a horizontal surface with surface tension and viscosity. Classical parabolic theory is not applicable to this type of equations, because, for example, of lack of comparison principle which is widely used in existence theory of second order parabolic equations.

The well known paper [4] by Bernis and Friedman initiated rigorous mathematical study of thin film equations. Namely, by using the energy-entropy method, in [4] the existence of nonnegative generalized weak solutions was proved for the special case $l\equiv 0$ of (1.3). These weak solutions were constructed as a limit of solutions of a regularized problem for a class of regular nonnegative initial data. The generalized weak solution introduced in [4] is in some sense “weaker” than a standard weak solution as not all integrals in the integral identity are defined on the whole domain of definition of the original problem. Existence of more regular (strong or entropy) solutions was shown in [3]. The classical thin-film equation, related to the case $l\not\equiv 0$ in our problem, has been well-studied over the last years. Multiple analytical results like: existence and regularity of solutions, convergence to a steady state, finite speed propagation and waiting time phenomenon were obtained in [15], [26], [27], [28]. However, to the best of our knowledge, the global asymptotic behaviour of solutions in the presence of absorption, was not studied for thin-film type equations before. This asymptotic behaviour, in infinite-dimensional dissipative systems, is associated with an existence of a global [29] or a trajectory [7] attractor. There is one fundamental complication for a straightforward application of a classical method of semigroups to (1.3). That is possible non-uniqueness of solutions to the corresponding initial boundary value problem (see [3] for simple examples of how, introduced by Bernis and Friedman, construction method applied to $h_t=-{(|h{|}^n{h}_{xxx})}_x$ can result in two different solutions arising from the same initial data). The attractor theory for complex systems without unique solutions was recently developed in [5], [6], [17], [19], [20], [21], [24], [32], [30] and this theory gave us a description of global asymptotic behaviour of weak and strong solutions for a wide class of evolution problems [14], [13], [16], [18], [31].

The main goals of this paper are: to analyse local and global solvability of the thin film equation with absorption, to obtain conditions for extra regularity of solutions, and to prove existence of a trajectory attractor for strong solutions.

The paper is structured as follows: in the second section we impose our main conditions on the parameters of the problem (1.3)–(1.5), we also give a definition and justify existence of weak and strong solutions (for existence of strong solutions see Theorem 1); in the third section we derive a priori estimates for the strong solutions of the problem; in the fourth section we prove the existence of a trajectory attractor (see Theorem 2) for the class of strong solutions that satisfy the corresponding a priori estimates.