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International Journal of Differential Equations
Volume 2012 (2012), Article ID 570283, 30 pages
http://dx.doi.org/10.1155/2012/570283
Research Article

Qualitative Analysis of Coating Flows on a Rotating Horizontal Cylinder

1School of Mathematical Sciences, Claremont Graduate University, 710 N. College Avenue, Claremont, CA 91711, USA
2School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK

Received 11 May 2012; Accepted 1 August 2012

Academic Editor: Sining Zheng

Copyright © 2012 Marina Chugunova and Roman M. Taranets. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider a nonlinear 4th-order degenerate parabolic partial differential equation that arises in modelling the dynamics of an incompressible thin liquid film on the outer surface of a rotating horizontal cylinder in the presence of gravity. The parameters involved determine a rich variety of qualitatively different flows. We obtain sufficient conditions for finite speed of support propagation and for waiting time phenomena by application of a new extension of Stampacchia's lemma for a system of functional equations.

1. Introduction

The time evolution of thickness of a viscous liquid film spreading over a solid surface under the action of the surface tension and gravity can be described by lubrication models [15]. These models approximate the full Navier-Stokes system that describes the motion of the liquid flow. Thin films play an increasingly important role in a wide range of applications, for example, packaging, barriers, membranes, sensors, semiconductor devices, and medical implants [68].

In this paper we consider the dynamics of a viscous incompressible thin fluid film on the outer surface of a horizontal circular cylinder that is rotating around its axis in the presence of a gravitational field. The motion of the liquid film is governed by four physical effects: viscosity, gravity, surface tension, and centrifugal forces. These are reflected in the parameters: 𝑅: the radius of the cylinder, 𝜔: its rate of rotation (assumed constant), 𝑔: the acceleration due to gravity, 𝜈: the kinematic viscosity, 𝜌: the fluid's density, and 𝜎: the surface tension. These parameters yield three independent dimensionless numbers: the Reynolds number Re=(𝑅2𝜔)/𝜈, 𝛾=𝑔/(𝑅𝜔2), and the Weber number We=(𝜌𝑅3𝜔2)/𝜎. The understanding of coating flow dynamics is important for industrial printing process where rotating cylinder transports the coating material in the form of liquid paint. The rotating thin fluid film can exhibit variety of different behaviour including: interesting pattern formations (“shark teeth” and “duck bill” patterns), fluid curtains, hydroplaning drops, and frontal avalanches [810]. As a result, the coating flow has been the subject of continuous study since the pioneering model was derived in 1977 by Moffatt (see [11]): 𝜕+𝜕𝜕𝑡1𝜕𝜃𝜔3𝑔𝜈𝑅3cos𝜃=0.(1.1) The surface tension and inertial effects were neglected in (1.1). Here (𝑥,𝑡) is the thickness of the fluid film, 𝜃 is a rotation angle, and 𝑡 is a time variable. The linear stability of rigidly rotating films on a rotating circular cylinder under three-dimensional disturbances was examined in [12, 13]. It was shown that the most unstable mode for thin film flows on the surface of a cylinder is the purely axial one that leads to so-called “ring instabilities”. During the past decade, coating and rimming problems attracted many researchers who analyzed different types of flow regime asymptotically [1417] and numerically [1820]. For a detailed review of a growing literature on different types of thin film flows please see [21] and references there in.

The coating flow is generated by viscous forces due to cylinder's surface motion relative to the fluid. There is no temperature gradient, hence the interface does not experience a shear stress. If the cylinder is fully coated there is only one free boundary where the liquid meets the surrounding air. Otherwise, there is also a free boundary (or contact line) where the air and liquid meet the cylinder's surface.

The asymptotic evolution equation for the thickness of the fluid film with the surface tension effect: 𝜕+𝜕𝜕𝑡1𝜕𝜃𝜔3𝑔𝜈𝑅31cos𝜃+3𝜎𝜌𝑅4𝜈3𝜕+𝜕𝜕𝜃3𝜕𝜃3=0,(1.2) was derived by Pukhnachev [22] in 1977. It is valid under the assumptions that the fluid film is thin 𝑅 and its slope is small (1/𝑅)(𝜕/𝜕𝜃)1. Later in 2009, taking into account inertial effects, Kelmanson [23] presented a more general model: 𝜕+𝜕𝜕𝑡1𝜕𝜃𝜔3𝑔𝜈𝑅31cos𝜃+3𝜎𝜌𝑅4𝜈3𝜕+𝜕𝜕𝜃3𝜕𝜃3+13𝜔2𝜌𝜈𝑅3𝜕𝜕𝜃=0.(1.3) He analyzed, asymptotically and numerically, diverse effects of inertia in both small- and large-surface-tension limits.

We should mention that all three lubrication approximation models described above were based on the assumption of the no-slip boundary condition. It is well known [24] that the combination of constant viscosity and no-slip boundary conditions at the liquid-solid interface yields a logarithmic divergence in the rate of dissipation at moving contact line, that is, an infinite energy is needed to make the droplet expand. The most common way to overcome this difficulty is to introduce effective slip conditions (see (2.1)) that indeed removes the force singularity at advancing contact lines (see [25]).

The main goal of our paper is to study waiting time phenomenon for the coating flows under an assumption of effective slip conditions, that is, we analyze (2.1) that is a modified version of (1.3). Our approach is based on now well-established nonlinear PDE analysis for degenerate higher order parabolic equations.

The sufficient conditions: 0(𝑥)𝐴|𝑥|4/𝑛 for 0<𝑛<2, |0𝑥(𝑥)|𝐵|𝑥|4/𝑛1 for 2𝑛<3, (where 𝐴 and 𝐵 are some positive constants) on nonnegative initial data, 0 for the occurrence of waiting time phenomena were derived by Dal Passo et al. [26] for the classic thin film equation: 𝑡+||||𝑛𝑥𝑥𝑥𝑥=0.(1.4) These results were based on an energy method developed in [27] for quasilinear parabolic equations. To the best of our knowledge, there is only one publication [28], where the waiting time phenomenon in the classic thin film equation (1.4) was discovered for 0(𝑥)|𝑥|𝛼 for 2<𝛼<4/𝑛. The result was obtained by means of matching asymptotic methods and was supported by numerous numerical simulations. For more general nonlinear degenerate parabolic equations with nonlinear lower order terms the waiting time phenomenon was analyzed in [2931].

It is well known [32] that the similarity solutions of the second order nonlinear parabolic equation: 𝑐𝑡=𝑐𝑚𝑐𝑥𝑥,𝑚>0,(1.5) subject to prescribing appropriate initial data, demonstrate the existence of a waiting-time phenomena before the free boundary moves. The comparison theorem, that is not applicable in our case, then enabled a number of results to be obtained about the existence and length of waiting times for general initial data. Our approach is completely different and based on local entropy/energy functional estimates.

We also analyze speed of support propagation and obtain an upper bound on it for the modified version of (1.3) (see (2.1)). The first finite speed results for nonnegative generalized solutions of the classic thin film equation (1.4) were obtained in [33, 34] for the case 0<𝑛<2 and 2𝑛<3, respectively. For more general types of thin film equations the finite speed of support propagation phenomenon was studied in [3539] (see also references there in).

The outline of our paper is as follows. We first prove for 𝑛>0 the long-time existence of a generalized weak solution and then prove that it can have an additional regularity in Section 2. In Sections 3 and 4 we show finite speed support propagation in the “slow” convection case (𝑛>1): for 1<𝑛<3 and waiting time phenomena for 1<𝑛<2, accordingly. The general strategy is to use an extension of Stampacchia's lemma for a system of functional equations (see Lemma  3.1 [26], where this extension is proved for a single equation and Lemma  A.2 in [37], where this extension is proved for systems in the homogeneous case). This result to our knowledge is new and might be of independent interest. We leave as an open problem the “fast” convection case (0<𝑛<1): finite speed of support propagation and sufficient conditions for waiting time phenomenon.

2. Existence and Regularity of Solutions

We are interested in the existence of nonnegative generalized weak solutions to the following initial-boundary value problem: (𝑃)𝑡+𝑓𝑎()0𝑥𝑥𝑥+𝑎1𝑥+𝑤𝑥𝑥=0in𝑄𝑇,𝜕𝑖𝜕𝑥𝑖𝜕(𝑎,𝑡)=𝑖𝜕𝑥𝑖(𝑎,𝑡)for𝑡>0,𝑖=0,3,(𝑥,0)=0(𝑥)0,(2.1) where 𝑓()=||𝑛,  =(𝑥,𝑡),  Ω=(𝑎,𝑎),  𝑄𝑇=(0,𝑇)×Ω,  𝑛>0,  𝑎0>0,  𝑎10, and 𝑤(𝑥,𝑡) such that 𝑤(𝑥,)𝑊1(0,𝑇)fora.e.𝑥Ω,𝑤(,𝑡)𝑊2[].(Ω)fora.e.𝑡0,𝑇(2.2) ote that (1.3) is a particular case of (2.1) that corresponds to 𝑛=3 and 𝑤(𝑥,𝑡)=cos(𝑥𝜔𝑡).

We consider a generalized weak solution in the following sense [40, 41].

Definition 2.1. A generalized weak solution of problem (𝑃) is a nonnegative function satisfying 𝐶1/2,1/8𝑥,𝑡𝑄𝑇𝐿0,𝑇;𝐻1(Ω),𝑡𝐿2𝐻0,𝑇;1(Ω),𝐶4,1𝑥,𝑡(𝒫),𝑎𝑓()0𝑥𝑥𝑥+𝑎1𝑥+𝑤𝑥𝐿2(𝒫),(2.3) where 𝒫={>0}. The solution satisfies (2.1) in the following sense: 𝑇0𝑡(,𝑡),𝜙𝑑𝑡𝒫𝑎𝑓()0𝑥𝑥𝑥+𝑎1𝑥+𝑤𝑥𝜙𝑥𝑑𝑥𝑑𝑡=0,(2.4) for all 𝜙𝐶1(𝑄𝑇)𝐶(𝑄𝑇) with 𝜙(𝑎,)=𝜙(𝑎,); (,𝑡)(,0)=0pointwise&stronglyin𝐿2[],𝜕(Ω)as𝑡0,(2.5)(𝑎,𝑡)=(𝑎,𝑡)𝑡0,𝑇𝑖𝜕𝑥𝑖𝜕(𝑎,𝑡)=𝑖𝜕𝑥𝑖(𝑎,𝑡),(2.6) for 𝑖=1,2,3 at all points of the lateral boundary where 0.

Because the second term of (2.4) has an integral over {>0} rather than over 𝑄𝑇, the generalized weak solution is “weaker” than a standard weak solution. Here, {>0} is short hand for {(𝑥,𝑡)𝑄𝑇(𝑥,𝑡)>0}. This short hand is used throughout: the time interval included in {>0} is to be inferred from the context it appears in.

A key object for proving additional properties of a generalized weak solution is an integral quantity introduced by Bernis and Friedman [42]: the “entropy” 𝐺0((𝑥,𝑡))𝑑𝑥. The function 𝐺0(𝑧) is defined by 𝐺0𝑧(𝑧)=2𝑛𝑧(2𝑛)(1𝑛)+𝑑𝑧+𝑐if𝑛1,2,𝑧ln𝑧𝑧+𝑒if𝑛=1,ln𝑧+𝑒+1if𝑛=2,(2.7) where 𝑑=1if1<𝑛<2,0otherwise,𝑐=(𝑛1)1/(1𝑛)2𝑛if1<𝑛<2,0otherwise.(2.8) By construction, 𝐺0 is a nonnegative convex function on [0,). For 1𝑛2, the linear part of 𝐺0 is chosen to ensure that 𝐺0 has a positive lower bound on [0,). Also in the statement of Theorem 2.2 we use an “𝛼-entropy”, 𝐺0(𝛼)((𝑥,𝑡))𝑑𝑥, where 𝐺0(𝛼)𝑧(𝑧)=𝑧ln𝑧𝑧+𝑒if𝛼=𝑛1,ln𝑧+𝑒𝑧+1if𝛼=𝑛2,2𝑛+𝛼(2𝑛+𝛼)(1𝑛+𝛼)+𝑑𝑧+𝑐otherwise,(2.9)𝑑=1if𝛼(𝑛2,𝑛1),0otherwise,𝑐=(𝑛1𝛼)1/(1+𝛼𝑛)2+𝛼𝑛if𝛼(𝑛2,𝑛1),0otherwise.(2.10)𝐺0(𝛼) is a nonnegative convex function on [0,). The linear part of 𝐺0(𝛼) is chosen to ensure that 𝐺0(𝛼) has a positive lower bound on [0,) if 𝑛2𝛼𝑛1. If 𝛼=0, the 𝛼-entropy is the same as the entropy (2.7).

Theorem 2.2. (a) (Existence). Let 𝑛>0 and the nonnegative initial data 0𝐻1(Ω), 0(𝑎)=0(𝑎) satisfy Ω𝐺00𝑑𝑥<.(2.11) Then for any time 0<𝑇< there exists a nonnegative generalized weak solution, , on 𝑄𝑇 in the sense of the Definition 2.1. Furthermore, 𝐿20,𝑇;𝐻2.(Ω)(2.12) Let 0(1𝑇)=2Ω𝑎02𝑥(𝑥,𝑇)𝑎12(𝑥,𝑇)2𝑤(𝑥,𝑇)(𝑥,𝑇)𝑑𝑥(2.13) then the weak solution satisfies 0(𝑇)+{>0}𝑛𝑎0𝑥𝑥𝑥+𝑎1𝑥+𝑤𝑥2𝑑𝑥𝑑𝑡0(0)𝑄𝑇𝑤𝑡𝑑𝑥𝑑𝑡.(2.14) (b) (Regularity). If the initial data also satisfies Ω𝐺0(𝛼)0𝑑𝑥<,(2.15) for some 1/2<𝛼<1,  𝛼0 then the nonnegative generalized weak solution has the extra regularity (𝛼+2)/2𝐿2(0,𝑇;𝐻2(Ω)) and (𝛼+2)/4𝐿4(0,𝑇;𝑊14(Ω)).

The theorem above was proved earlier in [41] for the case 𝑛=3 only. We note that the analogue of Theorem 4.2 in [42] also holds: there exists a nonnegative weak solution with the integral formulation 𝑇0𝑡(,𝑡),𝜙𝑑𝑡+𝑎0𝑄𝑇𝑛𝑛1𝑥𝑥𝑥𝜙𝑥+𝑛𝑥𝑥𝜙𝑥𝑥𝑑𝑥𝑑𝑡𝑄𝑇𝑛𝑎1𝑥+𝑤𝑥𝜙𝑥𝑑𝑥𝑑𝑡=0.(2.16) If initial data satisfy finite 𝛼-entropy condition, that is, 𝐺0(𝛼)(0)𝑑𝑥< then one can prove existence of nonnegative solutions with some additional regularity properties and use an integral formulation [43] to define them that is similar to that of (2.16) except that the second integral is replaced by the results of one more integration by parts (there are no 𝑥𝑥𝑥 terms). It is worth to mention that for the case 0<𝑛<2 the finite entropy assumption in Theorem 2.2 can be omitted because it does not impose any restriction on nonnegative initial data. One needs to impose finite entropy and finite 𝛼-entropy conditions on initial data if 𝑛2 only.

2.1. Regularized Problem

Given 𝛿,  𝜀>0, a regularized parabolic problem, similar to one that was studied by Bernis and Friedman [42] can be written as:

(𝑃𝛿,𝜖)𝑡+𝑓𝛿𝜀𝑎()0𝑥𝑥𝑥+𝑎1𝑥+𝑤𝑥𝑥𝜕=0,(2.17)𝑖𝜕𝑥𝑖𝜕(𝑎,𝑡)=𝑖𝜕𝑥𝑖(𝑎,𝑡)for𝑡>0,𝑖=0,3,(2.18)(𝑥,0)=0,𝜀(𝑥),(2.19) where 𝑓𝛿𝜀(𝑧)=𝑓𝜀(𝑧)+𝛿=|𝑧|4|𝑧|4𝑛+𝜀+𝛿𝑧1,𝛿>0,𝜀>0.(2.20) The 𝛿>0 in (2.20) makes the problem (2.17) regular (i.e., uniformly parabolic). The parameter 𝜀 is an approximating parameter which has the effect of increasing the degeneracy from 𝑓()||𝑛 to 𝑓𝜀()4. The nonnegative initial data, 0, is approximated via 0+𝜀𝜃0,𝜀𝐶4+𝛾2(Ω)forsome0<𝜃<5,𝜕𝑖0,𝜀𝜕𝑥𝑖𝜕(𝑎)=𝑖0,𝜀𝜕𝑥𝑖(𝑎)for𝑖=0,3,0,𝜀0stronglyin𝐻1(Ω)as𝜀0.(2.21) The 𝜀 term in (2.21) “lifts” the initial data so that they are smoothing from 𝐻1(Ω) to 𝐶4+𝛾(Ω). By Eĭdel’man [44, Theorem 6.3, p.302], the regularized problem has a unique classical solution 𝛿𝜀𝐶4+𝛾,1+𝛾/4𝑥,𝑡(Ω×[0,𝜏𝛿𝜀]) for some time 𝜏𝛿𝜀>0. For any fixed value of 𝛿 and 𝜀, by Eĭdel’man [44, Theorem 9.3, p.316] if one can prove a uniform in time a priori bound |𝛿𝜀(𝑥,𝑡)|𝐴𝛿𝜀< for some longer time interval [0,𝑇loc,𝛿𝜀]  (𝑇loc,𝛿𝜀>𝜏𝛿𝜀) and for all 𝑥Ω then Schauder-type interior estimates [44, Corollary 2, p.213] imply that the solution 𝛿𝜀 can be continued in time to be in 𝐶4+𝛾,1+𝛾/4𝑥,𝑡(Ω×[0,𝑇loc,𝛿𝜀]).

Although the solution 𝛿𝜀 is initially positive, there is no guarantee that it will remain nonnegative. The goal is to take 𝛿0,  𝜀0 in such a way that (1)  𝑇loc,𝛿𝜀𝑇loc>0, (2) the solutions 𝛿𝜀 converge to a (nonnegative) limit, , which is a generalized weak solution, and (3)   inherits certain a priori bounds. This is done by proving various a priori estimates for 𝛿𝜀 that are uniform in 𝛿 and 𝜀 and hold on a time interval [0,𝑇loc] that is independent of 𝛿 and 𝜀. As a result, {𝛿𝜀} will be a uniformly bounded and equicontinuous (in the 𝐶1/2,1/8𝑥,𝑡 norm) family of functions in Ω×[0,𝑇loc]. Taking 𝛿0 will result in a family of functions {𝜀} that are classical, positive, unique solutions to the regularized problem with 𝛿=0. Taking 𝜀0 will then result in the desired generalized weak solution . This last step is where the possibility of nonunique weak solutions arise; see [40] for simple examples of how such constructions applied to 𝑡=(||𝑛𝑥𝑥𝑥)𝑥 can result in two different solutions arising from the same initial data.

2.2. A Priori Estimates

Our first task is to derive a priori estimates for classical solutions of (2.17)–(2.21). The lemmas given in this section are proved in the Section 4.

We use an integral quantity based on a function 𝐺𝛿𝜀 chosen such that 𝐺𝛿𝜀1(𝑧)=𝑓𝛿𝜀(𝑧),𝐺𝛿𝜀(𝑧)0.(2.22) This is analogous to the “entropy” function first introduced by Bernis and Friedman [42].

Lemma 2.3. Let 0𝜀 satisfy (2.21) and be built from a nonnegative function 0 that satisfies the hypotheses of Theorem 2.2. Then there exist 𝛿0>0, 𝜀0>0 and time 𝑇loc>0 such that if 𝛿[0,𝛿0), 𝜀[0,𝜀0), and 𝛿𝜀 is a solution of the problem (2.17)–(2.21) with initial data 0𝜀, then for any 𝑇[0,𝑇loc] the following inequalities: Ω2𝛿𝜀,𝑥(𝑥,𝑇)+2𝑐1𝑎0𝐺𝛿𝜀𝛿𝜀(𝑥,𝑇)𝑑𝑥+𝑎0𝑄𝑇𝑓𝛿𝜀𝛿𝜀2𝛿𝜀,𝑥𝑥𝑥𝑑𝑥𝑑𝑡𝐾1<,(2.23)Ω𝐺𝛿𝜀𝛿𝜀(𝑥,𝑇)𝑑𝑥+𝑎0𝑄𝑇2𝛿𝜀,𝑥𝑥𝑑𝑥𝑑𝑡𝐾2<(2.24) hold. The energy 𝛿𝜀(𝑡) (see (2.13)) satisfies 𝛿𝜀(𝑇)+𝑄𝑇𝑓𝛿𝜀𝛿𝜀𝑎0𝛿𝜀,𝑥𝑥𝑥+𝑎1𝛿𝜀,𝑥+𝑤𝑥2𝑑𝑥𝑑𝑡=𝛿𝜀(0)𝑄𝑇𝛿𝜀𝑤𝑡𝑑𝑥𝑑𝑡.(2.25) The time 𝑇loc and the constants 𝐾𝑖 are independent of 𝛿 and 𝜀.

The proof of existence of 𝛿0,  𝜀0,  𝑇loc,  𝐾1, and 𝐾2 is constructive; how to find them and what quantities determine them are shown with details in Section 4.

Lemma 2.3 yields uniform-in-𝛿-and-𝜀 bounds for 2𝛿𝜀,𝑥, 𝐺𝛿𝜀(𝛿𝜀), 2𝛿𝜀,𝑥𝑥, and 𝑓𝛿𝜀(𝛿𝜀)2𝛿𝜀,𝑥𝑥𝑥. However, these bounds are found in a different manner than in earlier work for the equation 𝑡=(||𝑛𝑥𝑥𝑥)𝑥, for example. Although the inequality (2.24) is unchanged, the inequality (2.23) has an extra term involving 𝐺𝛿𝜀. In the proof, this term was introduced to control additional, lower-order terms. This idea of a “blended” 𝑥2-entropy bound was first introduced by Shishkov and Taranets for long-wave stable thin film equations with convection [30].

The final a priori bounds for positive, classical solutions use the following functions, parameterized by 𝛼 for 𝛼{2,3}, 𝐺𝜀(𝛼)(𝑧)=𝐺0(𝛼)𝑧(𝑧)+𝜀𝛼2𝐺(𝛼3)(𝛼2)𝜀(𝛼)(𝑧)=𝑧𝛼𝑓𝜀,(𝑧)(2.26) where 𝐺0(𝛼) is given by (2.9). In the following lemma, we restrict ourselves to the case 𝛼[1/2,1]; note that 𝐺𝜀(𝛼)(𝑧)0 for such 𝛼.

Lemma 2.4. Assume 𝜀0 and 𝑇loc are from Lemma 2.3, 𝛿=0, and 𝜀[0,𝜀0). Assume 𝛼[1/2,1] and that 𝜀 is a positive, classical solution of the problem (2.17)–(2.21) with initial data 0,𝜀 satisfying Lemma 2.3. If the initial data 0,𝜀 is built from 0 which also satisfies Ω𝐺0(𝛼)0(𝑥)𝑑𝑥<(2.27) then there exists 𝐾4 such that Ω2𝜀,𝑥(𝑥,𝑇)+𝐺𝜀(𝛼)𝜀(𝑥,𝑇)𝑑𝑥+𝑄𝑇𝛽𝛼𝜀2𝜀,𝑥𝑥+𝛾𝜀𝛼24𝜀,𝑥𝑑𝑥𝑑𝑡𝐾4<(2.28) holds for all 𝑇[0,𝑇loc] and 𝐾4 is independent of 𝜀 and is determined by 𝛼,  𝜀0,  𝑎0,  𝑎1,  𝑤𝑥,  Ω and 0. Here 𝑎𝛽=0𝑎if0𝛼1,01+2𝛼14(1𝛼)if2𝑎𝛼<0,𝛾=0𝛼(1𝛼)6𝑎if0𝛼1,0(1+2𝛼)(1𝛼)136f2𝛼<0.(2.29) Furthermore, if 𝛼(1/2,1){0} then 𝜀(𝛼+2)/2𝜀(0,𝜀0)𝐿20,𝑇loc;𝐻2,(Ω)𝜀(𝛼+2)/4𝜀(0,𝜀0)𝐿40,𝑇loc;𝑊1,4(Ω)(2.30) are uniformly bounded.

The 𝛼-entropy, 𝐺0(𝛼)()𝑑𝑥, was first introduced for 𝛼=1/2 in [45] and an a priori bound like that of Lemma 2.4 and regularity results like those of Theorem 2.2 were found simultaneously and independently in [40, 43].

The proof of existence starts from a construction of a classical solution 𝛿𝜀 on [0,𝑇loc] that satisfies the hypotheses of Lemma 2.3 if 𝛿(0,𝛿0) and 𝜀(0,𝜀0). Taking the regularizing parameter, 𝛿, to zero, one proves that there is a limit 𝜀 and that 𝜀 is a generalized weak solution. After that additional nonlinear estimates are required to analyze properties of the limit 𝜀; specifically to show that it is strictly positive, classical, and unique. Hence, the a priori bounds given by Lemmas 2.3 and 2.4 are applicable to 𝜀. This allows us to take the approximating parameter, 𝜀, to zero and to construct the desired nonnegative generalized weak solution of Theorems 2.2 (see, e.g., [41]).

2.3. Long-Time Existence of Solutions

Lemma 2.5. Let be a generalized solution of Theorem 2.2. Then 𝑎04,𝑇loc2𝐻1(Ω)0(0)+𝐾5+𝐾6𝑇loc,(2.31) where 0(0) is defined in (2.13), 𝑀=0, and 𝐾5=𝑤2𝑀+63𝑎0+𝑎13/2𝑎0𝑀2+𝑎0+𝑎12𝑀2||Ω||,𝐾6=𝑤𝑡𝑀.(2.32)

Proof of Lemma 2.5. By (2.13), 𝑎02Ω2𝑥(𝑥,𝑇)𝑑𝑥0(𝑎𝑇)+12Ω2(𝑥,𝑇)𝑑𝑥+Ω(𝑥,𝑇)𝑤(𝑥,𝑇)𝑑𝑥𝑄𝑇𝑤𝑡𝑑𝑥𝑑𝑡.(2.33) The linear-in-time bound (2.14) on 0(𝑇loc) then implies 𝑎02,𝑇loc2𝐻10(𝑎0)+0+𝑎12Ω2𝑑𝑥+𝑤+𝑤𝑡𝑇𝑀.(2.34) Using the estimate (see [41, Lemma 4.1, page 1837]) 2𝐿2(Ω)62/3𝑀4/3Ω2𝑥𝑑𝑥1/3+𝑀2||Ω||,(2.35) and Young's inequality: 𝑎0+𝑎12Ω2𝑎𝑑𝑥0+𝑎1262/3𝑀4/3Ω2𝑥𝑑𝑥1/3+𝑀2||Ω||𝑎04Ω2𝑥𝑥,𝑇loc2𝑑𝑥+63𝑎0+𝑎13/2𝑎0𝑀2+𝑎0+𝑎12𝑀2||Ω||.(2.36) Using this in (2.34), the desired bound (2.31) follows immediately.

This 𝐻1-estimate will be used to extend the short-time existence of a solution to the long-time existence result of Theorem 2.2 (see [41, Proof of Theorem 3, page 1838]).

3. Finite Speed of Support Propagation

Theorem 3.1. Let 1<𝑛<3. Assume 0 is nonnegative, 0𝐻1(Ω) and supp0(𝑟0,𝑟0)Ω. Then the solution of Theorem 2.2 has finite speed of support propagation, that is, there exists a continuous nondecreasing function Γ(𝑇),  Γ(0)=0 such that supp(𝑇,)(𝑟0Γ(𝑇),𝑟0+Γ(𝑇))Ω for all 𝑇𝑇0=Γ1(𝑎𝑟0).

In the following theorem, we find the explicit upper bounds of the Γ(𝑇) for a solution of the corresponding Cauchy problem with a compactly supported nonnegative initial data 0𝐻1(1). Note that the definition of generalized weak solution of the Cauchy problem is as Definition 2.1 except that Ω is replaced by 1 and the relation (2.6) is dropped. Using Lemma 2.5, we can show that the upper estimate of Γ(𝑇) from Theorem 3.1 is independent on Ω therefore the solution from Theorem 2.2 can be extended to be identically zero for |𝑥|>𝑟0Γ(𝑇) and thus is a solution on the line for all 𝑇𝑇0. Performing a similar procedure in [𝑇0,2𝑇0],,[𝑚𝑇0,(𝑚+1)𝑇0],, we obtain a compactly supported nonnegative solution of the Cauchy problem for all 𝑇0 and Theorem 2.2 holds with Ω=1.

Theorem 3.2. Let 1<𝑛<3. Assume 0 is nonnegative, 0𝐻1(1), supp0(𝑟0,𝑟0) and is a solution of the Cauchy problem. Then the following estimates:Γ(𝑇)𝐷1(𝑇1/(𝑛+4)+𝑇5/(𝑛+4)) for all 𝑇>0 if 1<𝑛<2,Γ(𝑇)𝐷2𝑇1/(𝑛+4) for small enough time if 2𝑛<3, are valid. Here the constants 𝐷𝑖 depend on the parameters problem and initial data only.

3.1. Proof of Theorem 3.1 for the Case 1<𝑛<2

The following lemma contains the local entropy estimate. The proof of Lemma 3.3 is similar to (A.16), (A.29), therefore it is omitted.

Lemma 3.3. Let 𝜁𝐶1,2𝑡,𝑥(𝑄𝑇) such that supp𝜁Ω, (𝜁4)=0 on 𝜕Ω, and 𝜁4(𝑎,𝑡)=𝜁4(𝑎,𝑡). Assume that 1/2<𝛼<1, and 𝛼0. Then there exist constants 𝐶𝑖(𝑖=1,2,3) dependent on 𝑛,  𝑚,  𝛼,  𝑎0, and 𝑎1, independent of Ω, such that for all 0<𝑇<Ω𝜁4(𝑥,𝑇)𝐺0(𝛼)((𝑥,𝑇))𝑑𝑥𝑄𝑇𝜁4𝑡𝐺0(𝛼)()𝑑𝑥𝑑𝑡+𝐶1𝑄𝑇(𝛼+2)/22𝑥𝑥𝜁4𝑑𝑥𝑑𝑡Ω𝜁4(𝑥,0)𝐺0(𝛼)0𝑑𝑥+𝐶2𝑄𝑇𝛼+2𝜁4+𝜁4𝑥+𝜁2𝜁2𝑥𝑥+𝜁2𝜁2𝑥+𝜁3||𝜁𝑥𝑥||𝑑𝑥𝑑𝑡+𝐶3𝑄𝑇𝛼+1||𝜁3||||𝜁𝑥||+𝜁4𝑑𝑥𝑑𝑡.(3.1)

Let 0<𝑛<2, and let supp0(𝑟0,𝑟0)Ω. For an arbitrary 𝑠(0,𝑎𝑟0) and 𝛿>0 we consider the families of sets Ω(𝑠)=Ω𝑟0𝑠,𝑟0+𝑠,𝑄𝑇(𝑠)=(0,𝑇)×Ω(𝑠).(3.2) We introduce a nonnegative cutoff function 𝜂(𝜏) from the space 𝐶2(1) with the following properties: 𝜏𝜂(𝜏)=0if𝜏0,2(32𝜏)if0<𝜏<1,1if𝜏1.(3.3) Next we introduce our main cut-off functions 𝜂𝑠,𝛿(𝑥)𝐶2(Ω) such that 0𝜂𝑠,𝛿(𝑥)1 for all 𝑥Ω and possess the following properties: 𝜂𝑠,𝛿𝑟(𝑥)=𝜂|𝑥|0+𝑠𝛿=||𝜂1,𝑥Ω(𝑠+𝛿),0,𝑥ΩΩ(𝑠),𝑠,𝛿𝑥||3𝛿,||𝜂𝑠,𝛿𝑥𝑥||6𝛿2,(3.4) for all 𝑠>0,  𝛿>0𝑟0+𝑠+𝛿<𝑎. Choosing 𝜁4(𝑥,𝑡)=𝜂𝑠,𝛿(𝑥)𝑒𝑡/𝑇, from (3.1) we arrive at Ω(𝑠+𝛿)𝛼𝑛+2(1𝑇)𝑑𝑥+𝑇𝑄𝑇(𝑠+𝛿)𝛼𝑛+2𝑑𝑥𝑑𝑡+𝐶𝑄𝑇(𝑠+𝛿)(𝛼+2)/22𝑥𝑥𝐶𝑑𝑥𝑑𝑡𝛿4𝑄𝑇(𝑠)𝛼+2𝐶𝑑𝑥𝑑𝑡+𝛿𝑄𝑇(𝑠)𝛼+1𝑑𝑥𝑑𝑡=𝐶2𝑖=1𝛿𝛼𝑖𝑄𝑇(𝑠)𝜉𝑖,(3.5) for all 𝑠(0,𝑎𝑟0), where (𝑛1)+<𝛼<1 and 0<𝛿<1 is enough small. We apply the Nirenberg-Gagliardo interpolation inequality (see Lemma B.2) in the region Ω(𝑠+𝛿) to a function 𝑣=(𝛼+2)/2 with 𝑎=(2𝜉𝑖)/(𝛼+2),  𝑏=(2(𝛼𝑛+2))/(𝛼+2),  𝑑=2,  𝑖=0,  𝑗=2, and 𝜃𝑖=((𝛼+2)(𝜉𝑖𝛼+𝑛2))/(𝜉𝑖(4𝛼3𝑛+8)) under the conditions: 𝛼𝑛+2<𝜉𝑖for𝑖=1,2.(3.6) Integrating the resulted inequalities with respect to time and taking into account (3.5), we arrive at the following relations: 𝑄𝑇(𝑠+𝛿)𝜉𝑖𝐶𝑇1(𝜃𝑖𝜉𝑖)/(𝛼+2)2𝑖=1𝛿𝛼𝑖𝑄𝑇(𝑠)𝜉𝑖1+𝜈𝑖+𝐶𝑇2𝑖=1𝛿𝛼𝑖𝑄𝑇(𝑠)𝜉𝑖𝜉𝑖/(𝛼𝑛+2),(3.7) where 𝜈𝑖=(4(𝜉𝑖𝛼+𝑛2))/(4𝛼3𝑛+8). These inequalities are true provided that 𝜃𝑖𝜉𝑖𝛼+2<1𝜉𝑖<5𝛼4𝑛+10for𝑖=1,2.(3.8) Simple calculations show that inequalities (3.6) and (3.8) hold with some (𝑛1)+<𝛼<1 if and only if 1<𝑛<2. The finite speed of propagations follows from (3.7) by applying Lemma B.3 with 𝑠1=0. Hence, supp(𝑇,)𝑟0Γ(𝑇),𝑟0+Γ(𝑇)Ωforall𝑇𝑇0,𝑇0,(3.9) where 𝑇0=Γ1(𝑎𝑟0).

3.2. Proof of Theorem 3.2 for the Case 1<𝑛<2

We can repeat the previous procedure from Section 3.1 for Ω(𝑠)=1(𝑟0𝑠,𝑟0+𝑠) and we obtain 𝐺𝑖(𝑠+𝛿)=𝑄𝑇(𝑠+𝛿)𝜉𝑖𝐶𝑇1(𝜃𝑖𝜉𝑖)/(𝛼+2)2𝑖=1𝛿𝛼𝑖𝑄𝑇(𝑠)𝜉𝑖1+𝜈𝑖,(3.10) instead of (3.7), and 𝑇Γ(𝑇)=𝐶(1(𝜃1𝜉1)/(𝛼+2))(1+𝜈2)𝑇(1(𝜃2𝜉2)/(𝛼+2))𝜈1(1+𝜈1)(𝐺(0))𝜈11/(4(1+𝜈1)(1+𝜈2))𝑇+𝐶(1(𝜃2𝜉2)/(𝛼+2))(1+𝜈1)𝑇(1(𝜃1𝜉1)/(𝛼+2))𝜈2(1+𝜈2)(𝐺(0))𝜈21/((1+𝜈1)(1+𝜈2)),(3.11) where 𝑇𝐺(0)=𝐶(1(𝜃1𝜉1)/(𝛼+2))(1+𝜈2)𝐺2(0)1+𝜈1+𝑇(1(𝜃2𝜉2)/(𝛼+2))(1+𝜈1)𝐺1(0)1+𝜈2.(3.12) Now we need to estimate 𝐺(0). With that end in view, we obtain the following estimates: 𝐺𝑖(0)𝐶1𝐶2+𝐶3𝑇(𝜉𝑖1)/(𝛼+5)𝑇1(𝜉𝑖1)/(𝛼+5),𝑖=1,2,(3.13) where 1<𝜉𝑖<𝛼+6, and 𝐶𝑖 depends on initial data only. Really, applying the Nirenberg-Gagliardo interpolation inequality (see Lemma B.2) in Ω=1 to a function 𝑣=(𝛼+2)/2 with 𝑎=(2𝜉𝑖)/(𝛼+2),  𝑏=2/(𝛼+2),  𝑑=2,  𝑖=0,  𝑗=2, and ̃𝜃𝑖=((𝛼+2)(𝜉𝑖1))/(𝜉𝑖(𝛼+5)) under the condition 𝜉𝑖>1, we deduce that 1𝜉𝑖𝑐0(2(3𝜉𝑖1+𝛼+2))/((𝛼+2)(𝛼+5))1(𝛼+2)/22𝑥𝑥𝑑𝑥(𝜉𝑖1)/(𝛼+5).(3.14) Integrating (3.14) with respect to time and taking into account the Hölder inequality ((𝜉𝑖1)/(𝛼+5)<1𝜉𝑖<𝛼+6), we arrive at the following relations: 𝑄𝑇𝜉𝑖𝑐0(2(3𝜉𝑖1+𝛼+2))/((𝛼+2)(𝛼+5))𝑇1(𝜉𝑖1)/(𝛼+5)𝑄𝑇(𝛼+2)/22𝑥𝑥𝑑𝑥(𝜉𝑖1)/(𝛼+5).(3.15) From (3.15), due to (A.16) (as 𝜀0) and (2.31), we find (3.13).

Inserting (3.13) into (3.12), we obtain after straightforward computations that 𝑇Γ(𝑇)𝐶1/(𝑛+4)+𝑇5/(𝑛+4)forall𝑇0.(3.16)

3.3. Proof of Theorem 3.1 for the Case 4/3<𝑛<3

The following lemma contains the local energy estimate. The proof of Lemma 3.4 is Appendix A.

Lemma 3.4. Let 𝑛(1/2,3) and 𝛽>(1𝑛)/3. Let 𝜁𝐶2(Ω) such that supp𝜁 in Ω and (𝜁6)=0 on 𝜕Ω, and 𝜁(𝑎)=𝜁(𝑎). Then there exist constants 𝐶𝑖(𝑖=1,3) dependent on 𝑛,  𝑚,  𝑎0, and 𝑎1, independent of Ω and 𝜀, such that for any 0<𝑇<Ω𝜁62𝑥(𝑥,𝑇)𝑑𝑥+Ω𝜁4𝛽+1(𝑇)𝑑𝑥+𝐶1𝑄𝑇𝜁6(𝑛+2)/22𝑥𝑥𝑥𝑑𝑥𝑑𝑡Ω𝜁620(𝑥)𝑑𝑥+Ω𝜁40𝛽+1𝑑𝑥+𝐶2𝑄𝑇𝑛+2𝜁6+𝜁6𝑥+𝜁3||𝜁𝑥𝑥||3𝑑𝑥𝑑𝑡+𝐶3𝑄𝑇𝜒{𝜁>0}𝑛+3𝛽1+𝑛𝜁6𝑑𝑥𝑑𝑡,(3.17)Ω𝜁62𝑥(𝑥,𝑇)𝑑𝑥+𝐶1𝑄𝑇𝜁6(𝑛+2)/22𝑥𝑥𝑥𝑑𝑥𝑑𝑡Ω𝜁620(𝑥)𝑑𝑥+𝐶2𝑄𝑇𝑛+2𝜁6+𝜁6𝑥+𝜁3||𝜁𝑥𝑥||3𝑑𝑥𝑑𝑡+𝐶3𝑄𝑇𝑛𝜁6𝑑𝑥𝑑𝑡.(3.18)

Let 𝜂𝑠,𝛿(𝑥) be denoted by (3.4). Setting 𝜁6(𝑥)=𝜂𝑠,𝛿(𝑥) into (3.17), after simple transformations, we obtain Ω(𝑠+𝛿)2𝑥(𝑥,𝑇)𝑑𝑥+Ω(𝑠+𝛿)𝛽+1(𝑇)𝑑𝑥+𝐶𝑄𝑇(𝑠+𝛿)(𝑛+2)/22𝑥𝑥𝑥𝐶𝑑𝑥𝑑𝑡𝛿6𝑄𝑇(𝑠)𝑛+2𝑑𝑥𝑑𝑡+𝐶𝑄𝑇(𝑠)𝑛+3𝛽1+𝑛𝑑𝑥𝑑𝑡=𝐶3𝑖=1𝛿𝛼𝑖𝑄𝑇(𝑠)𝜉𝑖,(3.19) for all for all 𝑠(0,𝑎𝑟0),𝛿>0𝑟0+𝑠+𝛿<𝑎. We apply the Nirenberg-Gagliardo interpolation inequality (see Lemma B.2) in the region Ω(𝑠+𝛿) to a function 𝑣=(𝑛+2)/2 with 𝑎=(2𝜉𝑖)/(𝑛+2),  𝑏=(2(𝛽+1))/(𝑛+2),  𝑑=2,  𝑖=0,  𝑗=3, and 𝜃𝑖=((𝑛+2)(𝜉𝑖𝛽1))/(𝜉𝑖(𝑛+5𝛽+7)) under the conditions: 𝛽<𝜉𝑖1for𝑖=1,3.(3.20) Integrating the resulted inequalities with respect to time and taking into account (3.19), we arrive at the following relations: 𝑄𝑇(𝑠+𝛿)𝜉𝑖𝐶𝑇1(𝜃𝑖𝜉𝑖)/(𝑛+2)3𝑖=1𝛿𝛼𝑖𝑄𝑇(𝑠)𝜉𝑖1+𝜈𝑖+𝐶𝑇3𝑖=1𝛿𝛼𝑖𝑄𝑇(𝑠)𝜉𝑖𝜉𝑖/(𝛽+1),(3.21) where 𝜈𝑖=(6(𝜉𝑖𝛽1))/(𝑛+5𝛽+7). These inequalities are true provided that 𝜃𝑖𝜉𝑖𝜉𝑛+2<1𝛽>𝑖𝑛86for𝑖=1,3.(3.22) Simple calculations show that inequalities (3.20) and (3.22) hold with some 𝛽((2𝑛)/2,𝑛1) if and only if 4/3<𝑛<3. Since all integrals on the right-hand sides of (3.21) vanish as 𝑇0, the finite speed of propagations follows from (3.21) by applying Lemma B.3 with 𝑠1=0 and sufficiently small 𝑇. Hence, supp(𝑇,)𝑟0Γ(𝑇),𝑟0+Γ(𝑇)Ωforall𝑇0𝑇𝑇0.(3.23)

3.4. Proof of Theorem 3.2 for the Case 4/3<𝑛<3

Suppose that Ω(𝑠)=1{𝑥|𝑥|<𝑠},  𝑄𝑇(𝑠)=(0,𝑇)×Ω(𝑠) for all 𝑠>𝑟0, supp0(𝑟0,𝑟0), and Γ(𝑇)=𝑟(𝑇)𝑟0. Since the time interval is small, we can assume that 𝑟(𝑇)<2𝑟0. Hence, for all 𝑠(𝑟0,2𝑟0), we can take (up to regularization) 𝜁=(|𝑥|𝑠)+ in (3.18). As a result, we obtain 12Ω(𝑠)(|𝑥|𝑠)6+2𝑥𝑑𝑥+𝛿6𝐶1𝑄𝑇(𝑠+𝛿)(𝑛+2)/22𝑥𝑥𝑥𝑑𝑥𝑑𝑡𝐶4𝑄𝑇(𝑠)𝑛+2+(𝑟(𝑇)𝑠)6+𝑛𝑑𝑥𝑑𝑡,(3.24) for all 𝑇𝑇0,  𝑠(𝑟0,2𝑟0). Using the Hardy type inequality Ω(𝑠)(|𝑥|𝑠)𝛼+𝑓2𝑑𝑥𝐶0Ω(𝑠)(|𝑥|𝑠)+𝛼+2𝑓2𝑥𝑑𝑥,(3.25) where 𝐶0=4/(𝛼+1)2 and 𝛼>1, we deduce that Ω(𝑠+𝛿)𝑑𝑥Ω(𝑠+𝛿)(|𝑥|𝑠)4+2𝑑𝑥1/2Ω(𝑠+𝛿)(|𝑥|𝑠)+4𝑑𝑥1/2𝐶03𝛿31/2Ω(𝑠)(|𝑥|𝑠)6+2𝑥𝑑𝑥1/2,(3.26) whence Ω(𝑠+𝛿)𝑑𝑥2𝐶03𝛿3Ω(𝑠)(|𝑥|𝑠)6+2𝑥𝑑𝑥,(3.27) for all 𝛿>0, 𝑠(𝑟0,2𝑟0). Substituting (3.27) in (3.24), we get 32𝐶0𝛿3sup𝑡Ω(𝑠+𝛿)𝑑𝑥2+𝐶1𝑄𝑇(𝑠+𝛿)(𝑛+2)/22𝑥𝑥𝑥𝐶𝑑𝑥𝑑𝑡4𝛿6𝑄𝑇(𝑠)𝑛+2+Γ6(𝑇)𝑛𝑑𝑥𝑑𝑡,(3.28) for all 𝑇𝑇0,  𝑠(𝑟0,2𝑟0). By the Nirenberg-Gagliardo, Hölder and Young inequalities, after simple transformations, for 𝜖𝑖>0, we have 𝐶4𝛿6𝑄𝑇(𝑠)𝑛+2𝑑𝑥𝑑𝑡𝜖1𝑄𝑇(𝑠)(𝑛+2)/22𝑥𝑥𝑥𝐶𝜖𝑑𝑥𝑑𝑡+1𝛿𝑛+7𝑇0Ω(𝑠)𝑑𝑥𝑛+2𝐶𝑑𝑡,4Γ6(𝑇)𝛿6𝑄𝑇(𝑠)𝑛𝑑𝑥𝑑𝑡𝜖2𝑄𝑇(𝑠)(𝑛+2)/22𝑥𝑥𝑥𝜖𝑑𝑥𝑑𝑡+𝐶2Γ(𝑇)𝛿3(𝑛+7)/4𝑇0Ω(𝑠)𝑑𝑥(3𝑛+1)/4𝑑𝑡.(3.29) Substituting the estimates (3.29) in (3.28) and making the standard iterative procedure for small enough 0<𝜖𝑖<1, we arrive at the inequality 32𝐶0sup𝑡Ω(𝑠+𝛿)𝑑𝑥2+𝐶5𝛿3𝑄𝑇(𝑠+𝛿)(𝑛+2)/22𝑥𝑥𝑥𝑑𝑥𝑑𝑡𝐶62𝑖=1𝐺𝑇(𝑖)(𝑠)𝛿𝛼𝑖,(3.30) where    𝛼1=𝑛+4,  𝛼2=(3(𝑛+3))/4,    𝐺𝑇(1)(𝑠)=𝑇0(Ω(𝑠)𝑑𝑥)𝑛+2𝑑𝑡,    𝐺𝑇(2)(𝑠)=Γ(3(𝑛+7))/4(𝑇)𝑇0(Ω(𝑠)𝑑𝑥)(3𝑛+1)/4𝑑𝑡. Thus, (3.30) yields 𝐺𝑇(𝑖)(𝑠+𝛿)𝐶7𝑇Γ𝜇𝑖(𝑇)2𝑖=1𝐺𝑇(𝑖)(𝑠)𝛿𝛼𝑖𝛽𝑖,(3.31) for all 𝑠(𝑟0,2𝑟0) and 0<𝛿<𝑠, where 𝜇1=0,  𝜇2=(3(𝑛+7))/4,  𝛽1=(𝑛+2)/2,  𝛽2=(3𝑛+1)/8. By Lemma B.3, from (3.31) we find that 𝐺𝑇(𝑖)(𝑠0)=0, where Γ(𝑇)𝑠0(𝑇)=𝐶8(𝑇1/𝛼1+𝑇1/𝛼2Γ𝜇2/𝛼2(𝑇)). As 𝜇2/𝛼2=(𝑛+7)/(𝑛+3)>1 for any 𝑇𝑇0, we have Γ(𝑇)𝐶9𝑇1/(𝑛+4).

4. Waiting Time Phenomenon

Let Ω(𝑠)={𝑥𝑥𝑠} for all 𝑠1, and 𝐡0(𝑠)=Ω(𝑠)0𝛼𝑛+2(𝑥)𝑑𝑥=0𝑠0,(4.1) where (𝑛1)+<𝛼<1. Let us assume that the function 𝐡0(𝑠) satisfies the flatness conditions. Namely, for every 𝑠𝑠0<𝑠<0 the following estimate: 𝐡0(𝑠)𝜒max(𝑠)1+(4(𝛼𝑛+2))/𝑛,(𝑠)1+(43𝑛+4(𝛼𝑛+2))/(4(𝑛1))=𝜒(𝑠)1+(4(𝛼𝑛+2))/𝑛4for3𝑛<2,𝜒(𝑠)1+(43𝑛+4(𝛼𝑛+2))/(4(𝑛1))4for1<𝑛<3,(4.2) is valid.

Theorem 4.1. Let 1<𝑛<2. Assume 0 is nonnegative, 0𝐻1(1) and meas{Ω(𝑠)supp0}= for all 𝑠0, that is, the condition (4.1) is valid, and the flatness condition (4.2) also holds.
Then for the solution of Theorem 2.2 (with Ω=1) there exists the time 𝑇=𝑇(𝜒)>0 depending on the known parameters only such that supp(𝑡,)Ω(0)=0<𝑡𝑇,(4.3) where 𝜒 is the constant from the flatness condition. Note, that 𝑇+ as 𝜒0.

Remark 4.2. Let the initial data 0𝐶(1) satisfy the following properties:(1) if 1<𝑛<4/3 then we suppose that sup𝑥Ω(𝑠)0(𝑥)𝜒(𝑠)(43𝑛+4(𝛼𝑛+2))/(4(𝑛1)(𝛼𝑛+2))forsome𝛼(𝑛1)+;,1(4.4) (2) if 4/3𝑛<2 then we suppose that sup𝑥Ω(𝑠)0(𝑥)𝜒(𝑠)4/𝑛.
These assumptions on the initial data are sufficient for the validity of flatness condition (4.2) and guarantee the appearance of the WTP, that is, the validation of property (4.3).

Remark 4.3. Note that due to Lemma 2.5 we have the estimate 𝛼+2(𝑥,𝑡)𝐶(1+𝑡)𝛼+1(𝑥,𝑡). Therefore, using this inequality in (3.1) with Ω=1, we could also obtain the waiting time phenomenon by the application of Theorem 2.1 from [46] with 𝑤=(𝛼+2)/2,  𝑙=𝑘=𝑝=2,  𝑞=(2(𝛼𝑛+2))/(𝛼+2), and 𝑠=(2(𝛼+1))/(𝛼+2).

Proof of Theorem 4.1. Similar to (3.10) for Ω(𝑠)={𝑥𝑥𝑠} and we obtain 𝐺𝑖(𝑠+𝛿)=𝑄𝑇(𝑠+𝛿)𝜉𝑖𝐾𝑇1(𝜃𝑖𝜉𝑖)/(𝛼+2)2𝑘=1𝛿𝛼𝑘𝐺𝑘(𝑠)+𝐡0(𝑠)1+𝜈𝑖.(4.5) Let us check that all conditions of Lemma B.4 are satisfied. We denote by 𝐺max(𝑠)=max𝑖=1,2𝑐02𝛽+12𝑘=1𝐺𝑘(𝑠)𝛽𝑘𝛽𝑖1(𝑠)1/(𝛼𝑖𝛽),𝑔max(𝑠)=max𝑖=1,22(𝛽+1)/(𝛼𝑖𝛽)22𝛽1𝑘=1𝐾𝑇1(𝜃𝑘𝜉𝑘)/(𝛼+2)𝛽𝑘𝛽𝑖/𝛼𝑖𝐡0(𝑠)(𝛽𝑖1)/𝛼𝑖,𝑐0=22𝛽1𝑘=1𝐾𝑇1(𝜃𝑘𝜉𝑘)/(𝛼+2)𝛽𝑘,𝛽𝑖=1+𝜈𝑖,𝛽=𝛽1𝛽2.(4.6) Taking 𝑠=2𝛿 in (4.5) and passing to the limit 𝛿, due to the boundedness of functions 𝐺𝑘(𝑠) and 𝐡0(𝑠), we deduce 𝐺𝑘()𝐾𝑇1(𝜃𝑘𝜉𝑘)/(𝛼+2)𝐡𝛽𝑘0().(4.7) This implies that the condition (i) of Lemma B.4 is fulfilled. Because of the assumption (4.2) on the function 𝐡0(𝑠), we can find 𝑇 such that the condition (ii) of Lemma B.4 is valid for all 𝑇[0,𝑇]. Here 𝑇=𝑇(𝜒) goes to infinity as 𝜒0. Hence, the application of Lemma B.4 ends the proof.

Appendices

A. Proofs of a Priori Estimates

The first observation is that the periodic boundary conditions imply that classical solutions of (2.17) conserve mass: Ω𝛿𝜀(𝑥,𝑡)𝑑𝑥=Ω0,𝜀(𝑥)𝑑𝑥=𝑀𝜀<forall𝑡>0.(A.1) Further, (2.21) implies 𝑀𝜀𝑀=0 as 𝜀0. Also, we will relate the 𝐿𝑝 norm of to the 𝐿𝑝 norm of its zero-mean part as follows: ||||||||𝑀(𝑥)(𝑥)𝜀Ω||||+𝑀𝜀Ω𝑝𝑝2𝑝1𝑣𝑝𝑝+2||Ω||𝑝1𝑀𝑝𝜀,(A.2) where 𝑣=𝑀𝜀/Ω. We will use the Poincaré inequality which holds for any zero-mean function in 𝐻1(Ω)𝑣𝑝𝑝𝑏1𝑣𝑥𝑝𝑝1𝑝<,𝑏1=||Ω||𝑝𝑝.(A.3) Also used will be an interpolation inequality [47, Theorem 2.2, page 62] for functions of zero mean in 𝐻1(Ω): 𝑣𝑝𝑝𝑏2𝑣𝑥2𝑎𝑝𝑣𝑟(1𝑎)𝑝,(A.4) where 𝑟1,  𝑝𝑟,  𝑎=(1/𝑟1/𝑝)/(1/𝑟+1/2),  𝑏2=(1+𝑟/2)𝑎𝑝. It follows that for any zero-mean function 𝑣 in 𝐻1(Ω)𝑣𝑝𝑝𝑏3𝑣𝑥𝑝2,𝑝𝑝𝑏4𝑥𝑝2+𝑏5𝑀𝑝𝜀,(A.5) where 𝑏3=𝑏1||Ω||(2𝑝)/𝑝𝑏if1𝑝21(𝑝+2)/2𝑏2𝑏if2<𝑝<,4=2𝑝1𝑏3,𝑏5=2||Ω||𝑝1.(A.6) To see that (A.5) holds, consider two cases. If 1𝑝<2, then by (A.3), 𝑣𝑝 is controlled by 𝑣𝑥𝑝. By the Hölder inequality, 𝑣𝑥𝑝 is then controlled by 𝑣𝑥2. If 𝑝>2 then by (A.4), 𝑣𝑝 is controlled by 𝑣𝑥𝑎2𝑣21𝑎 where 𝑎=1/21/𝑝. By the Poincaré inequality, 𝑣21𝑎 is controlled by 𝑣𝑥21𝑎.

Proof of Lemma 2.3. In the following, we denote the solution 𝛿𝜀 by whenever there is no chance of confusion.
To prove the bound (2.23) one starts by multiplying (2.17) by 𝑥𝑥, integrating over 𝑄𝑇, and using the periodic boundary conditions (2.18) yields 12Ω2𝑥(𝑥,𝑇)𝑑𝑥+𝑎0𝑄𝑇𝑓𝛿𝜀()2𝑥𝑥𝑥1𝑑𝑥𝑑𝑡=2Ω20𝜀,𝑥(𝑥)𝑑𝑥𝑎1𝑄𝑇𝑓𝛿𝜀()𝑥𝑥𝑥𝑥𝑑𝑥𝑑𝑡𝑄𝑇𝑓𝛿𝜀()𝑤𝑥𝑥𝑥𝑥𝑑𝑥𝑑𝑡.(A.7) By Cauchy and Young inequalities, due to (A.3)–(A.5), it follows from (A.7) that 12Ω2𝑥(𝑎𝑥,𝑇)𝑑𝑥+02𝑄𝑇𝑓𝛿𝜀()2𝑥𝑥𝑥1𝑑𝑥𝑑𝑡2Ω20𝜀,𝑥𝑑𝑥+𝑐1𝑄𝑇2𝑥𝑥𝑑𝑥𝑑𝑡+𝑐2𝑇0max1,Ω2𝑥𝑑𝑥𝜅1𝑑𝑡,(A.8) where 𝜅1=max{𝑛,3},  𝑐1=(𝑎21/4𝑎0)𝑏2,  𝑐2=(𝑎21/2𝑎0)𝑏4+(𝑎21/2𝑎0)𝑏5𝑀𝜀2𝑛+(𝑎21/4𝑎0)𝑏2+(𝑎21/𝑎0)𝛿 + sup𝑡𝑇((𝑤𝑥22/𝑎0)𝛿+(𝑤𝑥2/𝑎0)𝑏4+(𝑤𝑥2/𝑎0)𝑏5𝑀𝑛𝜀). Multiplying (2.17) by 𝐺𝛿𝜀(), integrating over 𝑄𝑇, and using the periodic boundary conditions (2.18), we obtain Ω𝐺𝛿𝜀((𝑥,𝑇))𝑑𝑥+𝑎0𝑄𝑇2𝑥𝑥𝑑𝑥𝑑𝑡Ω𝐺𝛿𝜀0𝜀𝑑𝑥+𝑐3𝑇0max1,Ω2𝑥(𝑥,𝑡)𝑑𝑥𝑑𝑡,(A.9) where 𝑐3=𝑎1+sup𝑡𝑇𝑤𝑥2. Further, from (A.8) and (A.9) we find Ω2𝑥𝑑𝑥+2𝑐1𝑎0Ω𝐺𝛿𝜀((𝑥,𝑇))𝑑𝑥+𝑎0𝑄𝑇𝑓𝛿𝜀()2𝑥𝑥𝑥𝑑𝑥𝑑𝑡Ω20𝜀,𝑥𝑑𝑥+2𝑐1𝑎0Ω𝐺𝛿𝜀0𝜀𝑑𝑥+𝑐4𝑇0max1,Ω2𝑥(𝑥,𝑡)𝑑𝑥𝜅1𝑑𝑡,(A.10) where 𝑐4=2𝑐1𝑐3/𝑎0+2𝑐2. Applying the nonlinear Grönwall lemma [48] to 𝑣(𝑇)𝑣(0)+𝑐4𝑇0max{1,𝑣𝜅1(𝑡)}𝑑𝑡 with 𝑣(𝑡)=(2𝑥(𝑥,𝑡)+2𝑐1/𝑎0𝐺𝛿𝜀((𝑥,𝑡)))𝑑𝑥 yields Ω2𝑥(𝑐𝑥,𝑡)+21𝑎0𝐺𝛿𝜀((𝑥,𝑡))𝑑𝑥21/(𝜅11)max1,Ω20𝜀,𝑥(𝑥)+2𝑐1𝑎0𝐺𝛿𝜀0𝜀(𝑥)𝑑𝑥=𝐾𝛿𝜀<(A.11) for all 𝑡[0,𝑇𝛿𝜀,loc], where 𝑇𝛿𝜀,loc1=2𝑐4𝜅11min1,Ω20𝜀,𝑥(𝑥)+2𝑐1𝑎0𝐺𝛿𝜀0𝜀(𝑥)𝑑𝑥(𝜅11).(A.12) Using the 𝛿0,  𝜀0 convergence of the initial data and the choice of 𝜃(0,2/5) (see (2.21)) as well as the assumption that the initial data 0 has finite entropy (2.11), the times 𝑇𝛿𝜀,loc converge to a positive limit and the upper bound 𝐾 in (A.11) can be taken finite and independent of 𝛿 and 𝜀 for 𝛿 and 𝜀 sufficiently small. Therefore there exists 𝛿0>0 and 𝜀0>0 and 𝐾 such that the bound (A.11) holds for all 0𝛿<𝛿0 and 0𝜀<𝜀0 with 𝐾 replacing 𝐾𝛿𝜀 and for all 0𝑡𝑇loc9=10lim𝜀0,𝛿0𝑇𝛿𝜀,loc.(A.13)
Using the uniform bound on 2𝑥 that (A.11) provides, one can find a uniform-in-𝛿-and-𝜀 bound for the right-hand-side of (A.10) yielding the desired a priori bound (2.23). Similarly, one can find a uniform-in-𝛿-and-𝜀 bound for the right-hand-side of (A.9) yielding the desired a priori bound (2.24). The time 𝑇loc and the constant 𝐾 are determined by 𝛿0,  𝜀0,  𝑎0,  𝑎1,  sup𝑡𝑇𝑤𝑥2, sup𝑡𝑇𝑤𝑥, 0, 0𝑥2, and 𝐺0(0).
To prove the bound (2.25), multiply (2.17) by 𝑎0𝑥𝑥𝑎1𝑤, integrate over 𝑄𝑇, integrate by parts, use the periodic boundary conditions (2.18) to find (2.25).

Proof of Lemma 2.4. In the following, we denote the positive, classical solution 𝜀 by whenever there is no chance of confusion.
Multiplying (2.17) by (𝐺𝜀(𝛼)()), integrating over 𝑄𝑇, taking 𝛿0, and using the periodic boundary conditions (2.18) yield Ω𝐺