An Existence Theorem for a Nonlocal Global Pandemic Model for Insect-Borne Diseases

Abstract

We construct and analyze a nonlocal global pandemic model that comprises a system of two nonlocal integrodifferential equations (functional differential equations) and an ordinary differential equation. This model was constructed by considering a spherical coordinate transformation of a previously established epidemiology model that can be applied to insect-borne diseases, like yellow fever. This transformation amounts to a nonlocal boundary value problem on the unit sphere and therefore can be interpreted as a global pandemic model for insect-borne diseases. We ultimately show that a weak solution to the weak formulation of this model exists using a fixed point argument, which calls upon the construction of a weak formulation and the existence and uniqueness of an auxiliary problem.

1. Introduction

In [1], Cannon and Galiffa analyzed the nonlocal model: −𝛼¹₀𝑢(𝑡)d𝑡𝑢^′′(𝑥)()𝑢(0)(1)=𝑓,𝑥∈0,1=𝑎,𝑢=𝑏, (1) with 𝛼=𝛼(𝑞) being a positive function of 𝑞 defined over −∞<𝑞<∞, 𝑓(𝑥) defined over 0≤𝑥≤1, and 𝑎 and 𝑏 real constants. The analysis contained in [1] included both existence and uniqueness theorems for the analytic solution, 𝑢, to (1) and a discretization, as well as the construction and implementation of a finite-difference procedure. This model was a special case of [2], which has applications to thermodynamics.

The following homogeneous extension of model (1) was considered in [3]: −𝛼¹₀𝑢(𝑡)d𝑡𝑢^′′(𝑥)+[𝑢(𝑥)]^2𝑛+1(),𝑢(0)(1)=0,𝑥∈0,1=𝑎,𝑢=𝑏. (2) As in (1), the existence and uniqueness of the analytic solution, 𝑢, to (2), as well as a discretization, were established and another finite-difference procedure was developed and executed; however, the analysis was significantly different than in [1].

Upon the completion of [3], the authors realized the potential for additional applications to physical problems via mathematical models consisting of variations of the nonlocal term 𝛼(∫¹₀𝑢(𝑡)𝑑𝑡). In particular, a model was constructed and analyzed in [4, 5] that can be applied to insect-borne diseases, for example, yellow fever, which is as follows: 𝑢_𝑡=𝑢_𝑥𝑥−𝛼^𝑏_𝑎𝑤()𝑣𝑥,𝑡d𝑥𝑢+𝛾𝑤,−∞<𝑥<∞,𝑡≤𝑇,_𝑡=𝑣_𝑥𝑥+𝛼^𝑏_𝑎𝑤()𝑤𝑥,𝑡d𝑥𝑢−𝛽𝑣,−∞<𝑥<∞,𝑡≤𝑇,_𝑡𝑢()(𝑥)𝑣()𝑤()=𝛽𝑣−𝛾𝑤,−∞<𝑥<∞,𝑡≤𝑇,𝑥,0=𝑓,−∞<𝑥<∞,𝑥,0=0,−∞<𝑥<∞,𝑥,0=0,−∞<𝑥<∞ (3) with 𝛽 and 𝛾 arbitrary positive constants.

We next give an overview of this model, which is based on [5]. Biologically, 𝛽 represents the rate at which mobile infecteds become immobile and 𝛾 represents the rate at which immobile infecteds recover. The unknowns 𝑢, 𝑣, and 𝑤, respectively, represent a susceptible class, a mobile infected class, and an immobile infected class. In addition, neither infected class is infective; that is, susceptibles become infected via contact with vectors (disease-spreading agents, e.g., Aedes aegypti mosquitoes), which was not modeled directly. The nonnegative function 𝛼(𝑞), 0≤𝑞<∞, represents the per capita rate at which susceptibles contract the disease in a disease-free population, which is decreasing with 𝑞. The argument ∫^𝑏_𝑎𝑤(𝑥,𝑡)d𝑥, representing the number of the population immobilized from the disease, when substituted into 𝛼 brings about a reduction in the rate of infection via actions taken at large and at the individual level, to reduce the risk of infections. Even more specifically, ∫^𝑏_𝑎𝑤(𝑥,𝑡)d𝑥 counts the number of immobile infecteds within a single fixed interval [𝑎,𝑏] and influences the rate 𝛼(𝑞) from a biological standpoint. For example, if all news stories about immobile infecteds concern individuals living only in region [𝑎,𝑏], then susceptibles will take precautions based only on the reported number ∫^𝑏_𝑎𝑤(𝑥,𝑡)d𝑥.

In this paper, we consider an extension of model (3) via a spherical coordinate transformation that results in the following model on the unit sphere 𝒮: 𝑢_𝑡=1𝜕sin𝜙𝜕𝜙sin𝜙𝜕𝑢𝜕𝜙+csc²𝜙𝜕²𝑢𝜕𝜃²−𝛼_Ω𝑤(],𝑣𝜙,𝜃,𝑡sin𝜙d𝜙d𝜃𝑢+𝛾𝑤,on𝒮×0,𝑇_𝑡=1𝜕sin𝜙𝜕𝜙sin𝜙𝜕𝑤𝜕𝜙+csc²𝜙𝜕²𝑣𝜕𝜃²+𝛼_Ω𝑤(],𝑤𝜙,𝜃,𝑡sin𝜙d𝜙d𝜃𝑢−𝛽𝑣,on𝒮×0,𝑇_𝑡(],𝑢𝑣𝑤=𝛽𝑣𝜙,𝜃,𝑡−𝛾𝑤𝜙,𝜃,𝑡,on𝒮×0,𝑇𝜙,𝜃,0=𝑓𝜙,𝜃on𝒮,𝜙,𝜃,0=0on𝒮,𝜙,𝜃,0=0onΩ⊂𝒮. (4) Physically, the system comprising (4) can be interpreted as a preliminary model for a global pandemic of an insect-borne disease. First, we note that the system of spherical coordinates is the natural and historical coordinate system for the surface of the Earth. Second, we note that diffusion has been used to describe epidemics in the past; compare [6–9] for a few of several such examples. Third, the system proposed herein is preliminary, since we take no account of the oceans, as to do so will require boundary conditions to be assigned to the coasts of the Earth's land masses. Fourth, spherical coordinates come into play for relatively short distances on the Earth's surface. For example, a Cartesian coordinate system (𝑥,𝑦,𝑧) with origin located at the John F. Kennedy Space Center will find the plane (𝑥,𝑦,0) approximately 20,000 feet over Tampa, FL. Fifth, we have made no attempt to model air travel, which can be handled by the introduction of spatially located sources and sinks. Sixth, we note that mosquitoes have been found alive in shipping containers filled with used tires shipped to the United States from Asia, as well as other locations, which provides a vector for tropical Asian diseases on the American continent (cf. http://www.epa.state.il.us/land/tires/mosquito-borne-illnesses.html, for one such example). Moreover, we also note that passengers entering Australia are sprayed with insecticide disinfectant before they can be allowed to deplane (http://abcnews.go.com/).

Finally, our interest here is to show that a solution exists to (4) in order to enable the study of the possible effect of the death toll of a disease on the dynamics of the epidemic itself. Thus, the system (4), which incorporates a coefficient containing a functional of part of the solution of the system, requires a nonlinear analysis, which we will outline later in this introduction.

Several papers have recently been published regarding the mathematical analysis of pandemics, as well as the construction of mathematical models comprising a system of differential equations applicable to such biological phenomena. Several of these papers influenced our current work and it is therefore worth briefly describing them before we further discuss (4).

For example, in [10] the authors explain how the utilization of antiviral drugs was intended to moderate the severity of a new strain of an influenza pandemic and that the success of these drugs was dependent upon the timely onset of therapy (within 48 hours) after the appearance of clinical symptoms. They discuss further that this requirement can be understood by a compartmental model that examines the density of infected individuals in terms of the time elapsed since the onset of symptoms. Therefore, based on this compartmental model, a system of delay-differential equations, with both discrete and disturbed delays, was constructed and analyzed. This system is as follows: 𝑑𝑆(𝑡)(𝑡)𝑄(𝑡),𝑑𝑑𝑡=−𝛽𝑆𝐸(𝑡)(𝑡)𝑄(𝑡)𝑑𝑡=𝛽𝑆−𝜇_𝐸𝐸(𝑡), (5) for 𝑡≥0, where susceptible and exposed classes are, respectively, denoted by 𝑆 and 𝐸, with 𝛽 representing the baseline transmission rate, 1/𝜇_𝐸 representing the incubation period, and 𝑄(𝑡) representing the force of infection yet to be formulated.

As a precaution to a potential H5N1 pandemic, the authors in [11] simulate a possible outbreak in Italy by analyzing the structure and the effect of including more specific details like heterogeneities and stochasticity. More specifically, the heterogeneity was with respect to age and spatial distribution, and the stochasticity regulated the epidemic dynamics when the number of the infected population is low. This resulted in the SEIR model: 𝑑𝑆𝑑𝑡^𝑝𝑖=−𝑆^𝑝_{𝑖𝑗,𝑞}𝛽^𝑝𝑞_𝑖𝑗𝐼^𝑞_𝑗𝑁^𝑞_𝑗,𝑑𝐼𝑑𝑡^𝑝𝑖=𝜈𝐸^𝑝_𝑖−𝛾𝐼^𝑝_𝑖,𝑑𝐸𝑑𝑡^𝑝𝑖=𝑆^𝑝_{𝑖𝑗,𝑞}𝛽^𝑝𝑞_𝑖𝑗𝐼^𝑞_𝑗𝑁^𝑞_𝑗−𝜈𝐸^𝑝_𝑖,𝑑𝑅𝑑𝑡^𝑝𝑖=𝐼^𝑝_𝑖, (6) with 𝑖,𝑗=1,…,12 and 𝑝,𝑞=1,…,20, where 1/𝜈 and 1/𝛾, respectively, represent the mean length of the latency and infectivity periods.

Other fairly recent papers related to the modeling of pandemics are worth mentioning here as well. For example, in [12] genetic algorithms were used in order to determine optimal vaccination strategies for pandemic influenza. Also, in [13] in order to analyze recurrent outbreaks of the avian H5N1 influenza virus in Asia, which constitute a global pandemic threat, the authors analyzed the great influenza pandemic of 1918 in Geneva, Switzerland, which is also referred to as the “Spanish flu.” Lastly, [14] shows how previous constraints regarding continuous space contact models describing a class of multitype epidemics can be removed in order to construct a general proof of the pandemic bonds for the multitype epidemic on the lattice ℤ^𝑁. For further interesting results related to recent studies of pandemics consider [15–17].

Our current model is quite different than the aforementioned ones, since it comprises two nonlocal integrodifferential equations, also often referred to as functional differential equations, and an ordinary differential equation. Moreover, in order to establish the existence of the weak solution to the weak formulation of (4), an interesting Hilbert space is constructed, as established in Section 3. In addition, several facets of the analysis needed in the study of Laplace's equation on the unit sphere are used throughout. We also emphasize that, to the very best of our knowledge, there are currently no published results related to nonlocal global pandemic models.

This paper is organized as follows. In Section 2, we derive a weak formulation of (4). Based on this analysis, an auxiliary problem is constructed in Section 3, from which we show that its solution exists and is unique. In Section 4, we prove the existence of a solution to the weak formulation of model (4), using the analysis of Section 3. We conclude the paper with Section 5, wherein we briefly discuss future considerations.

2. A Weak Formulation

We begin by multiplying each equation in (4) by sin𝜙, resulting in 𝑢_𝑡𝜕sin𝜙=𝜕𝜙sin𝜙𝜕𝑢𝜕𝜕𝜙+csc𝜙²𝑢𝜕𝜃²−𝛼_Ω𝑤(],𝜙,𝜃,𝑡sin𝜙d𝜙d𝜃𝑢sin𝜙+𝛾𝑤sin𝜙,on𝒮×0,𝑇𝑣_𝑡𝜕sin𝜙=𝜕𝜙sin𝜙𝜕𝑤𝜕𝜕𝜙+csc𝜙²𝑣𝜕𝜃²+𝛼_Ω𝑤(],𝜙,𝜃,𝑡sin𝜙d𝜙d𝜃𝑢sin𝜙−𝛽𝑣sin𝜙,on𝒮×0,𝑇𝑤_𝑡=𝛽𝑣𝜙,𝜃,𝑡−𝛾𝑤𝜙,𝜃,𝑡,on𝒮×(0,𝑇],𝑢𝑤𝜙,𝜃,0=𝑓𝜙,𝜃on𝒮,𝑣𝜙,𝜃,0=0on𝒮,𝜙,𝜃,0=0onΩ⊂𝒮. (7)(8)(9)(10) Moreover, 𝑢, 𝑣, and 𝑤, as well as 𝜕𝑢/𝜕𝜃, 𝜕𝑣/𝜕𝜃, and 𝜕𝑤/𝜕𝜃, are all continuous across the data line, and all of the derivatives with respect to 𝜙 of 𝑢, 𝑣, and 𝑤 exist for every direction 𝜃 when 𝜙=0,𝜋.

In considering a weak solution to (7)–(9), we need to establish appropriate test functions. Consider 𝑔(𝜙,𝜃,𝑡)∈𝐶¹ in 𝜙, 𝜃, and 𝑡 and let 𝑔 and 𝑔_𝜃 be continuous across the data line. Now multiply (7) by 𝑔(𝜙,𝜃,𝑡) and let 𝑔(𝜙,𝜃,𝑇)=0; then ^𝑇₀^𝜋_−𝜋^𝜋₀𝑔𝑢_𝑡sin𝜙d𝜙d𝜃d𝑡=−^𝜋_−𝜋^𝜋₀𝑔𝑓−𝜙,𝜃,0𝜙,𝜃sin𝜙d𝜙d𝜃_{𝑇0𝜋−𝜋𝜋0}𝑢𝑔_𝑡sin𝜙d𝜙d𝜃d𝑡. (11) The next term in (7) yields ^𝑇₀^𝜋_−𝜋^𝜋₀𝑔𝜕𝜕𝜙sin𝜙𝑢_𝜙=d𝜙d𝜃d𝑡_{𝑇0𝜋−𝜋}𝑔𝑢_𝜙sin𝜙∣^𝜙=𝜋_𝜙=0−d𝜃d𝑡_{𝑇0𝜋−𝜋𝜋0}𝑔𝑢_𝜙sin𝜙d𝜙d𝜃d𝑡. (12) We also have ^𝑇₀^𝜋₀^𝜋_−𝜋csc𝜙𝑔𝑢_𝜃𝜃d𝜃_𝜙=d𝜙d𝑡_𝑇0𝜋0csc𝜙𝑔𝑢_𝜃∣^𝜃=𝜋_𝜃=−𝜋−d𝜙d𝑡_{𝑇0𝜋0𝜋−𝜋}csc𝜙𝑔_𝜃𝑢_𝜃d𝜃d𝜙d𝑡. (13) Rearranging terms we obtain −^𝑇₀^𝜋_−𝜋^𝜋₀𝑢𝑔_𝑡−sin𝜙d𝜙d𝜃d𝑡_{𝑇0𝜋−𝜋𝜋0}𝑔_𝜙𝑢_𝜙+sin𝜙d𝜙d𝜃d𝑡_{𝑇0𝜋−𝜋𝜋0}csc𝜙𝑔_𝜃𝑢_𝜃=d𝜙d𝜃d𝑡,_{𝜋−𝜋𝜋0}𝑔𝑓−𝜙,𝜃,0𝜙,𝜃sin𝜙d𝜙d𝜃_{𝑇0𝜋−𝜋𝜋0}𝛼_Ω𝑤sin𝜙d𝜙d𝜃𝑔_𝜙𝑢sin𝜙d𝜙d𝜃d𝑡+𝛾^𝑇₀^𝜋_−𝜋^𝜋₀𝑔_𝜙𝑤sin𝜙d𝜙d𝜃, (14) for all 𝑔∈𝐶¹(𝑄_𝑇) and 𝑄_𝑇={(𝜙,𝜃,𝑡)∣0≤𝜙≤𝜋,−𝜋≤𝜃≤𝜋,0<𝑡≤𝑇}.

Now let 𝐷_𝑡∶={(𝜙,𝜃,𝑡)∣0≤𝜙≤𝜋,−𝜋≤𝜃≤𝜋}. For ease of notation we assign the following: _𝑄𝑇𝑢𝑔_𝑡sin𝜙d𝑄_𝑇∶=^𝑇₀^𝜋_−𝜋^𝜋₀𝑢𝑔_𝑡sin𝜙d𝜙d𝜃d𝑡,_𝐷𝑡𝑢𝑔_𝑡sin𝜙d𝐷_𝑡∶=^𝜋_−𝜋^𝜋₀𝑢𝑔_𝑡sin𝜙d𝜙d𝜃,_Ω𝑤dΩ∶=_Ω𝑤𝜙,𝜃,𝑡sin𝜙d𝜙d𝜃,whereΩ⊆𝒮. (15) We can then write the weak form of (7) as follows: −_𝑄𝑇𝑢𝑔_𝑡sin𝜙d𝑄_𝑇−_𝑄𝑇𝑢_𝜙𝑔_𝜙sin𝜙d𝑄_𝑇+_𝑄𝑇𝑢_𝜃𝑔_𝜃csc𝜙d𝑄_𝑇=_𝐷0𝑔𝑓sin𝜙d𝐷₀−_𝑄𝑇𝛼_Ω𝑤dΩ𝑢𝑔sin𝜙d𝑄_𝑇+𝛾_𝑄𝑇𝑔𝑤sin𝜙d𝑄_𝑇, (16) for all 𝑔∈𝐶¹(𝑄_𝑇) and 𝑔(𝜙,𝜃,𝑇)=0, or equivalently for all 𝑔∈𝑊^1,1, which is the closure of 𝐶¹(𝑄_𝑇) with 𝑔(𝜙,𝜃,𝑇)=0 in the norm 𝑔²_𝑊1,1∶=_𝑄𝑇𝑔²sin𝜙d𝑄_𝑇+_𝑄𝑇𝑔²_𝜃sin𝜙d𝑄_𝑇+_𝑄𝑇𝑔²_𝜃csc𝜙d𝑄_𝑇+_𝑄𝑇𝑔²_𝑡sin𝜙d𝑄_𝑇. (17) For the weak form of (8) we have −_𝑄𝑇𝑣𝑔_𝑡sin𝜙d𝑄_𝑇+_𝑄𝑇𝑣_𝜙𝑔_𝜙sin𝜙d𝑄_𝑇+_𝑄𝑇𝑣_𝜃𝑔_𝜃csc𝜙d𝑄_𝑇=_𝑄𝑇𝛼_Ω𝑤dΩ𝑢𝑔sin𝜙d𝑄_𝑇−𝛽_𝑄𝑇𝑣𝑔sin𝜙d𝑄_𝑇,∀𝑔∈𝑊^1,1. (18) On the other hand, the weak form of (9) is the pointwise integral representation for almost all (𝜙,𝜃,𝑡)∈𝑄_𝑇, which is 𝑤=𝛽^𝑡₀𝑣𝑒^{−𝛾(𝑡−𝜏)}d𝜏. (19) The integral representation (19) of 𝑤 is a nonlinear functional equation for 𝑤 as ∫_Ω𝑤dΩ occurs in the function 𝛼 in both (16) and (18) for 𝑢 and 𝑣.

3. The Auxiliary Problem

We replace ∫_Ω𝑤dΩ in the function 𝛼 in (16) and (18) with the continuous function 𝑞(𝑡) to yield the auxiliary problem of finding 𝑢(𝜙,𝜃,𝑡;𝑞), 𝑣(𝜙,𝜃,𝑡;𝑞), and 𝑤(𝜙,𝜃,𝑡;𝑞) satisfying −_𝑄𝑇𝑢𝑔_𝑡sin𝜙d𝑄_𝑇+_𝑄𝑇𝑢_𝜙𝑔_𝜙sin𝜙d𝑄_𝑇+_𝑄𝑇𝑢_𝜃𝑔_𝜃csc𝜙d𝑄_𝑇=_𝐷0𝑔𝑓sin𝜙d𝐷₀−_𝑄𝑇𝛼𝑞(𝑡)𝑢𝑔sin𝜙d𝑄_𝑇+𝛾_𝑄𝑇𝑔𝑤sin𝜙d𝑄_𝑇,∀𝑔∈𝑊^1,1,−_𝑄𝑇𝑣𝑔_𝑡sin𝜙d𝑄_𝑇+_𝑄𝑇𝑣_𝜙𝑔_𝜙sin𝜙d𝑄_𝑇+_𝑄𝑇𝑣_𝜃𝑔_𝜃csc𝜙d𝑄_𝑇=_𝑄𝑇𝛼𝑞(𝑡)𝑢𝑔sin𝜙𝑑𝑄_𝑇−𝛽_𝑄𝑇𝑣𝑔sin𝜙d𝑄_𝑇,∀𝑔∈𝑊^1,1,𝑤𝜙,𝜃,𝑡;𝑞=𝛽^𝑡₀𝑣𝜙,𝜃,𝜏;𝑞𝑒^{−𝛾(𝑡−𝜏)}d𝜏 (20) for almost all (𝜙,𝜃,𝑡)∈𝑄_𝑇. It should also be noted that we have already assumed that 𝑔 is periodic in 𝜃. Also, we can extend 𝑔 as an odd function in 𝜙 with −𝜋≤𝜙≤0 and regard it as periodic in 𝜙. Therefore, we can utilize the functions sin𝑛𝜙sin𝑚𝜃 and sin𝑛𝜙cos𝑚𝜃 in order to generate an indexed set of functions ℎ_𝑘(𝜙,𝜃) by first counting the functions sin𝑛𝜙cos𝑚𝜃 according to an orderly method of counting the ordered pairs (𝑛,𝑚) for 𝑛=1,2,… and 𝑚=0,1,2,… in the closure of the 𝑥-𝑦 plane via the diagonal pattern ()⟶()⟶()⟶()⟶()⟶()⟶()⟶()⟶()⟶()⟶()⟶()⟶()⟶()⟶()⟶()⟶()1,01,12,03,02,11,21,32,23,14,05,04,13,22,31,41,52,4⋯. (21) Next, take the number index assigned to each pair (𝑛,𝑚) and double it, yielding the ℎ_𝑘-terms for even integers. Then, for the odd integers, as we did above, count the functions sin𝑛𝜙cos𝑚𝜃 according to an orderly method of counting the ordered pairs (𝑛,𝑚) for 𝑛=1,2,… and 𝑚=1,2,… in the closure of the 𝑥-𝑦 plane via the diagonal pattern ()⟶()⟶()⟶()⟶()⟶()⟶()⟶()⟶()⟶()⟶()⟶()⟶()⟶()⟶()⟶()⟶()1,11,22,13,12,21,31,42,33,24,15,14,23,32,41,51,62,5⋯. (22) So, we have a list of ℎ_𝑘(𝜙,𝜃) for 𝑘=1,2,…. The first six of such are as follows: ℎ₁=sin𝜙sin(𝜃),ℎ₂𝜙,ℎ=sin₃=sin𝜙sin(2𝜃),ℎ₄𝜙(𝜃),ℎ=sincos₅=sin2𝜙sin(𝜃),ℎ₆=sin2𝜙,…. (23)

Since the integrals involving the ℎ²_𝑘-term, given by ^𝜋_−𝜋^𝜋₀sin²𝑛𝜙cos²𝑚𝜃sin𝜙d𝜙d𝜃,_{𝜋−𝜋𝜋0}sin²𝑛𝜙sin²𝑚𝜃sin𝜙d𝜙d𝜃, (24) converge, and the integrals involving the 𝜕ℎ²_𝑘/𝜕𝜙-term, which are ^𝜋_−𝜋^𝜋₀𝑛²cos²𝑛𝜙cos²𝑚𝜃sin𝜙d𝜙d𝜃,_{𝜋−𝜋𝜋0}𝑛²cos²𝑛𝜙sin²𝑚𝜃sin𝜙d𝜙d𝜃, (25) also converge, we then consider the integrals involving the 𝜕ℎ²_𝑘/𝜕𝜃-term below: ^𝜋_−𝜋^𝜋₀𝑚²csc𝜙sin²𝑛𝜙sin²𝑚𝜃d𝜙d𝜃,_{𝜋−𝜋𝜋0}𝑚²csc𝜙sin²𝑛𝜙cos²𝑚𝜃d𝜙d𝜃 (26) and remark that ^𝜋₀𝑚²csc𝜙sin²𝑛𝑛𝜙d𝜙<²𝑚²𝜋⁴8. (27) Hence, ^𝜋_−𝜋^𝜋₀ℎ²_𝑘sin𝜙d𝜙d𝜃+^𝜋_−𝜋^𝜋₀𝜕ℎ_𝑘𝜕𝜙²+sin𝜙d𝜙d𝜃_{𝜋−𝜋𝜋0}𝜕ℎ_𝑘𝜕𝜃²csc𝜙d𝜙d𝜃<∞, (28) for 𝑘=1,2,…. Thus, via the Gram-Schmidt process, we can generate a countable orthonormal basis for the Hilbert space ℒ²(𝐷₀) with the inner product 𝑓𝜙,𝜃,𝑔𝜙,𝜃_𝐷0=_𝐷0𝑓𝑔sin𝜙d𝐷₀+_𝐷0𝜕𝑓𝜕𝜙𝜕𝑔𝜕𝜙sin𝜙d𝐷₀+_𝐷0𝜕𝑓𝜕𝜃𝜕𝑔𝜕𝜃csc𝜙d𝐷₀, (29) where 𝐷₀={(𝜙,𝜃)∣0≤𝜙≤𝜋,−𝜋≤𝜃≤𝜋}. Henceforth, we denote the orthonormal basis formed from ℎ_𝑘(𝜙,𝜃) via the Gram-Schmidt process by {ℎ_𝑘(𝜙,𝜃)}.

Now, define 𝑢_𝑁𝜙,𝜃,𝑡=^𝑁_𝑘=1𝑎^𝑁_𝑘ℎ_𝑘,𝑣𝜙,𝜃_𝑁𝜙,𝜃,𝑡=^𝑁_𝑘=1𝑏^𝑁_𝑘ℎ_𝑘,𝑤𝜙,𝜃_𝑁𝜙,𝜃,𝑡=^𝑁_𝑘=1𝑐^𝑁_𝑘ℎ_𝑘𝜙,𝜃. (30) From there, we construct a system of ODEs as follows: _𝐷0𝑢_𝑛ℎ_𝑖sin𝜙d𝐷₀+_𝐷0𝜕𝑢_𝑁𝜕𝜙𝜕ℎ_𝑖𝜕𝜙sin𝜙d𝐷₀+_𝐷0𝜕𝑢_𝑁𝜕𝜃𝜕ℎ_𝑖𝜕𝜃csc𝜙d𝐷₀=−_𝐷0𝛼𝑞(𝑡)𝑢_𝑁ℎ_𝑖sin𝜙d𝐷₀+𝛾_𝐷0𝑤_𝑁ℎ_𝑖sin𝜙d𝐷₀,_𝐷0𝑣_𝑛ℎ_𝑖sin𝜙d𝐷₀+_𝐷0𝜕𝑣_𝑁𝜕𝜙𝜕ℎ_𝑖𝜕𝜙sin𝜙d𝐷₀+_𝐷0𝜕𝑣_𝑁𝜕𝜃𝜕ℎ_𝑖𝜕𝜃csc𝜙d𝐷₀=_𝐷0𝛼𝑞(𝑡)𝑢_𝑁ℎ_𝑖sin𝜙d𝐷₀−𝛽_𝐷0𝑣_𝑁ℎ_𝑖sin𝜙d𝐷₀,_𝐷0𝑤_𝑛ℎ_𝑖sin𝜙d𝐷₀=𝛽_𝐷0𝑣_𝑁ℎ_𝑖sin𝜙d𝐷₀−𝛾_𝐷0𝑤_𝑁ℎ_𝑖sin𝜙d𝐷₀,𝑎^𝑁_𝑖(0)=𝑓,ℎ_𝑖𝐷0,𝑏^𝑁_𝑖(0)=𝑐^𝑁_𝑖(0)=0,𝑖=1,2,…,𝑁. (31) The equations above become 𝑑𝑎^𝑁_𝑖(𝑡)+𝑑𝑡^𝑁_𝑙=1𝐴^𝑁_𝑖,𝑙𝑎^𝑁_𝑙(𝑡)=−𝛼𝑞(𝑡)𝑎^𝑁_𝑖(𝑡)+𝛾𝑐^𝑁_𝑖(𝑡),𝑑𝑏^𝑁_𝑖(𝑡)+𝑑𝑡^𝑁_𝑙=1𝐵^𝑁_𝑖,𝑙𝑏^𝑁_𝑙(𝑡)=𝛼𝑞(𝑡)𝑎^𝑁_𝑖(𝑡)−𝛽𝑏^𝑁_𝑖(𝑡),𝑑𝑐^𝑁_𝑖(𝑡)𝑑𝑡=𝛽𝑏^𝑁_𝑖(𝑡)−𝛾𝑐^𝑁_𝑖(𝑡),𝑎^𝑁_𝑖(0)=𝑓,ℎ_𝑖𝐷0,𝑏^𝑁_𝑖(0)=𝑐^𝑁_𝑖(0)=0,𝑖=1,2,…,𝑁. (32) As the above equations are linear, for each positive integer 𝑁 there exists a solution on [0,𝑇] since 𝛼(𝑞(𝑡)) is bounded for 𝑞(𝑡)≥0. Hence, for each 𝑁, 𝑢_𝑁, 𝑣_𝑁, and 𝑤_𝑁 are determined uniquely. We note that 𝑐^𝑁_𝑖(𝑡)=𝛽^𝑡₀𝑏^𝑁_𝑖(𝜏)𝑒^{𝛾(𝑡−𝜏)}d𝜏. (33) Upon multiplying each of the equations in the system (32) by the coefficient corresponding to the 𝑡-derivative in the equation and summing each equation over 𝑖=1,…,𝑁, we achieve 𝑑𝑢_𝑁𝑑𝑡,𝑢_𝑁𝐷0+_𝐷0𝜕𝑢_𝑁𝜕𝜙²sin𝜙d𝐷₀+_𝐷0𝜕𝑢_𝑁𝜕𝜃²csc𝜙d𝐷₀=−_𝐷0𝛼𝑞(𝑡)𝑢²_𝑁sin𝜙d𝐷₀+𝛾_𝐷0𝑤_𝑁𝑢_𝑁sin𝜙d𝐷₀,𝑑𝑣_𝑁𝑑𝑡,𝑣_𝑁𝐷0+_𝐷0𝜕𝑣_𝑁𝜕𝜙²sin𝜙d𝐷₀+_𝐷0𝜕𝑣_𝑁𝜕𝜃²csc𝜙d𝐷₀=_𝐷0𝛼𝑞(𝑡)𝑢_𝑁𝑣_𝑁sin𝜙d𝐷₀+𝛽_𝐷0𝑣²_𝑁sin𝜙d𝐷₀,𝑑𝑤_𝑁𝑑𝑡,𝑤_𝑁𝐷0=𝛽_𝐷0𝑣_𝑁𝑤_𝑁sin𝜙d𝐷₀−𝛾_𝐷0𝑤²_𝑁sin𝜙d𝐷₀ (34) and integrating each of the above equations from 0 to 𝑡 we obtain 12𝑢_𝑁(𝜙,𝜃,𝑡)²_𝐷0+_𝑄𝑡𝜕𝑢_𝑁𝜕𝜙²sin𝜙d𝑄_𝑡+_𝑄𝑡𝜕𝑢_𝑁𝜕𝜃²csc𝜙d𝑄_𝑡=12𝑢_𝑁𝜙,𝜃,0²_𝐷0−_𝑄𝑡𝛼𝑞(𝑡)𝑢²_𝑁sin𝜙d𝑄_𝑡+𝛾_𝑄𝑡𝑤_𝑁𝑢_𝑁sin𝜙d𝑄_𝑡,12𝑣_𝑁𝜙,𝜃,𝑡²_𝐷0+_𝑄𝑡𝜕𝑣_𝑁𝜕𝜙²sin𝜙d𝑄_𝑡+_𝑄𝑡𝜕𝑣_𝑁𝜕𝜃²csc𝜙d𝑄_𝑡=_𝑄𝑡𝛼𝑞(𝑡)𝑢_𝑁𝑣_𝑁sin𝜙d𝑄_𝑡−𝛽_𝑄𝑡𝑣²_𝑁sin𝜙d𝑄_𝑡,12𝑤_𝑁(𝜙,𝜃,𝑡)²_𝐷0=𝛽_𝑄𝑡𝑣_𝑁𝑤_𝑁sin𝜙d𝑄_𝑡−𝛾_𝑄𝑡𝑤²_𝑁sin𝜙d𝑄_𝑡. (35) Multiplying the above equations by 2, utilizing the inequality 2𝑎𝑏≤𝑎²+𝑏² on the right-hand sides of each equation, adding the inequalities, and dropping the 𝑄_𝑡 integrals on the left-hand side of the resulting inequalities, we have 𝐹(𝑡)≤𝑓²_𝐷0+𝐶₁^𝑡₀𝐹(𝜏)d𝜏, (36) where 𝐹(𝑡)∶=𝑢_𝑁(𝜙,𝜃,𝑡)²_𝐷0+𝑣_𝑁(𝜙,𝜃,𝑡)²_𝐷0+𝑤_𝑁(𝜙,𝜃,𝑡)²_𝐷0,𝐶₁=3‖𝛼‖_∞+3𝛽+2𝛾+1 (37) and ‖𝑢_𝑁(𝜙,𝜃,0)‖²_𝐷0≤‖𝑓‖²_𝐷0.

Then, by Gronwall’s lemma we see that 𝑦^′≤𝑓²_𝐷0+𝐶₁𝑦𝑑𝑑𝑡𝑦𝑒^−𝐶1𝑡≤𝑓²_𝐷0𝑒^−𝐶1𝑡𝑦(𝑡)≤𝑓²_𝐷0𝐶₁𝑒^𝐶1𝑡−1. (38) Therefore, 𝐹(𝑡)≤𝑓²_𝐷0𝑒^𝐶1𝑡. (39) In addition, we have 𝐹(𝑡)+2_𝑄𝑡𝜕𝑢_𝑁𝜕𝜙²sin𝜙d𝑄_𝑡+2_𝑄𝑡𝜕𝑢_𝑁𝜕𝜃²csc𝜙d𝑄_𝑡+2_𝑄𝑡𝜕𝑣_𝑁𝜕𝜙²sin𝜙d𝑄_𝑡+2_𝑄𝑡𝜕𝑣_𝑁𝜕𝜃²csc𝜙d𝑄_𝑡≤𝑓²_𝐷0+𝐶₁^𝑡₀𝐹(𝜏)𝑑𝜏 (40) and we observe that the entire left-hand side of the above inequality is also bounded by ‖𝑓‖²_𝐷0𝑒^𝐶1𝑡 for each 𝑡∈[0,𝑇] and so is each individual term on the left-hand side.

From there, we see that _𝑄𝑇𝑢²_𝑁sin𝜙d𝑄_𝑇,_𝑄𝑇𝑣²_𝑁sin𝜙d𝑄_𝑇,_𝑄𝑇𝑤²_𝑁sin𝜙d𝑄_𝑇 (41) are all bounded uniformly with respect to 𝑁. Likewise, 𝑢_𝑁 and 𝑣_𝑁 are also uniformly bounded in the Hilbert space 𝒱 with inner product ℎ,𝑔_𝒱=_𝑄𝑇ℎ𝑔sin𝜙d𝑄_𝑇+_𝑄𝑇ℎ_𝜙𝑔_𝜙sin𝜙d𝑄_𝑇+_𝑄𝑇ℎ_𝜃𝑔_𝜃csc𝜙d𝑄_𝑇, (42) where ℎ_𝜙, 𝑔_𝜙, ℎ_𝜃, and 𝑔_𝜃 are the first partial derivatives of ℎ and 𝑔, with respect to the subscripted variables.

Next, we take (33) and multiply it by ℎ_𝑖(𝜙,𝜃) and sum with 𝑖 from 1 to 𝑁 and obtain 𝑤_𝑁𝜙,𝜃,𝑡=𝛽^𝑡₀𝑣_𝑁𝜙,𝜃,𝜏𝑒^{−𝛾(𝑡−𝜏)}∀d𝜏,𝜙,𝜃,𝑡∈𝑄_𝑇. (43) Thus, by a Cantor diagonalization process, there exist sequences {𝑢_𝑁}, {𝑣_𝑁}, and {𝑤_𝑁}, which converge weakly in ℒ²(𝑄_𝑇) with {𝑢_𝑁} and {𝑣_𝑁} also converging weakly in 𝒱. Let 𝑢, 𝑣, and 𝑤 denote the weak limits and let 𝑔_𝑚𝜙,𝜃,𝑡=^𝑚_𝑖=1𝑒_𝑖(𝑡)ℎ_𝑖𝜙,𝜃 (44) denote an approximation in 𝑊₂^1,1(𝑄_𝑇) to 𝑔(𝜙,𝜃,𝑡)∈𝐶₁(𝑄_𝑇) with 𝑒_𝑖(𝑇)=0 for all 𝑖=1,2,…; that is, 𝑔_𝑚→𝑔 strongly in 𝑊₂^1,1(𝑄_𝑇) and 𝑔_𝑚(𝜙,𝜃,𝑇)=𝑔(𝜙,𝜃,𝑇)=0. Taking the system of ODEs and multiplying each by 𝑒_𝑖(𝑡) and summing each block of equations with respect to 𝑖 transform the system as follows: 𝜕𝑢_𝑁𝜕𝑡,𝑔_𝑚𝐷0+_𝐷0𝜕𝑢_𝑁𝜕𝜙𝜕𝑢_𝑁𝑔𝜕𝜙^𝑚sin𝜙d𝐷₀+_𝐷0𝜕𝑢_𝑁𝜕𝜃𝜕𝑢_𝑁𝑔𝜕𝜃^𝑚csc𝜙d𝐷₀=_𝐷0𝛼𝑞(𝑡)𝑢_𝑁𝑔_𝑚sin𝜙d𝐷₀+𝛾_𝐷0𝑤_𝑁𝑔_𝑚sin𝜙d𝐷₀,𝜕𝑣_𝑁𝜕𝑡,𝑔_𝑚𝐷0+_𝐷0𝜕𝑣_𝑁𝜕𝜙𝜕𝑣_𝑁𝑔𝜕𝜙^𝑚sin𝜙d𝐷₀+_𝐷0𝜕𝑣_𝑁𝜕𝜃𝜕𝑣_𝑁𝑔𝜕𝜃^𝑚csc𝜙d𝐷₀=_𝐷0𝛼𝑞(𝑡)𝑢_𝑁𝑔_𝑚sin𝜙d𝐷₀−𝛽_𝐷0𝑣_𝑁𝑔_𝑚sin𝜙d𝐷₀,_𝑄𝑇𝑤_𝑁−𝛽^𝑡₀𝑣_𝑁𝑒^{−𝛾(𝑡−𝜏)}d𝜏𝑔_𝑚d𝑄_𝑇=0. (45) Integrating with respect to 𝑡 from 0 to 𝑇 and performing integration by parts on the first term in each equation, we observe that 𝑢_𝑁, 𝑣_𝑁, and 𝑤_𝑁 satisfy the weak formulation for 𝑔_𝑚. Then holding 𝑔_𝑚 fixed and letting 𝑁→∞ we see that the weak limits 𝑢, 𝑣, and 𝑤 satisfy the original system (20) for each 𝑚. As 𝑔_𝑚→𝑔∈𝐶¹(𝑄_𝑡) in the norm of 𝑊₂^1,1(𝑄_𝑡), then 𝑢, 𝑣, and 𝑤 satisfy the weak formulation for all 𝑔∈𝐶¹(𝑄_𝑡)∩𝑊₂^1,1 and 𝑔(𝜙,𝜃,𝑇)=0. Thus, as 𝑔 is dense in 𝑊₂^1,1 the weak limits 𝑢, 𝑣, and 𝑤 together form a weak solution to the auxiliary problem (20).

Upon considering the energy inequality for parabolic equations (cf. [18]), it can be shown that any weak solution for 𝑢, 𝑣, and 𝑤 satisfies an energy inequality similar to the one for {𝑢_𝑁,𝑣_𝑁,𝑤_𝑁}. Hence, 𝑢, 𝑣, and 𝑤 satisfy the same bounds as 𝑢_𝑁, 𝑣_𝑁, and 𝑤_𝑁. Consequently, by the linearity of the system 𝑓(𝜙,𝜃)≡0, we have 𝑢=𝑣=𝑤=0 and thus, the weak solution is unique.

4. Existence of a Solution to the Auxiliary Problem

As the auxiliary problem possesses a unique weak solution for each continuous 𝑞(𝑡), we can now define a mapping ℱ∶𝐶([0,𝑇])→𝐶([0,𝑇]) via ℱ𝑞(𝑡)=_Ω𝑤𝜙,𝜃,𝑡;𝑞dΩ. (46) First, we show that (ℱ𝑞)(𝑡) is bounded and continuous on [0,𝑇]. From integrating the pointwise representation for 𝑤(𝜙,𝜃,𝑡;𝑞), we see that _Ω𝑤dΩ=𝛽_Ω^𝑡₀𝑣𝑒^{−𝛾(𝑡−𝜏)}d𝜏dΩ (47) and, taking into account the fact that 0<𝑒^{−𝛾(𝑡−𝜏)}≤1, we have _Ω𝑤dΩ≤𝛽^𝑡_0Ω|𝑣|dΩd𝜏=𝛽^𝑡_0Ω|𝑣|sin𝜙d𝜃d𝜙d𝜏≤𝛽^𝑡_0Ωsin𝜙d𝜃d𝜙d𝜏^1/2×^𝑡_0Ω𝑣²sin𝜙d𝜃d𝜙d𝜏^1/2≤𝛽𝑇^1/2[mes(Ω)]^1/2_𝑄𝑇𝑣²sin𝜙d𝑄_𝑇^1/2≤𝐶₂, (48) via the Schwartz inequality, where 𝐶₂ is a positive constant that depends on 𝐶₁ of Section 3, 𝑇, and mes(Ω).

For the continuity of (ℱ𝑞)(𝑡), we consider 0≤𝑡₁<𝑡₂≤𝑇, and thus _Ω𝑤𝜙,𝜃,𝑡₁;𝑞dΩ−_Ω𝑤𝜙,𝜃,𝑡₂;𝑞dΩ≤𝛽_Ω^𝑡1₀𝑣𝜙,𝜃,𝜏;𝑞𝑒^{−𝛾(𝑡1−𝜏)}d𝜏−𝛽^𝑡2₀𝑣𝜙,𝜃,𝜏;𝑞𝑒^{−𝛾(𝑡2−𝜏)}d𝜏dΩ≤𝛽_Ω^𝑡2_𝑡1𝑣𝑒^{−𝛾(𝑡2−𝜏)}d𝜏dΩ+𝛽_Ω^𝑡1₀𝑣𝑒^{−𝛾(𝑡1−𝜏)}−𝑒^{−𝛾(𝑡2−𝜏)}d𝜏dΩ≤𝛽_Ω^𝑡2_𝑡1|𝑣|d𝜏dΩ+𝛽_Ω^𝑡1₀|𝑣|𝑒^{−𝛾(𝑡1−𝜏)}1−𝑒^{𝛾(𝑡1−𝜏)−𝛾(𝑡2−𝜏)}d𝜏dΩ≤𝛽_Ω^𝑡2_𝑡1|𝑣|d𝜏dΩ+𝛽_Ω^𝑡1₀|𝑣|1−𝑒^{−𝛾(𝑡2−𝑡1)}d𝜏dΩ≤𝛽_Ω^𝑡2_𝑡1|𝑣|d𝜏dΩ+𝛽_Ω^𝑡1₀|𝑣|𝛾𝑡₂−𝑡₁d𝜏dΩ≤𝐶₃𝑡₂−𝑡₁^1/2+𝐶₄𝑡₂−𝑡₁, (49) where we used the fact that 0<𝑒^{−𝛾(𝑡−𝜏)}≤1, estimated 1−𝑒^{−𝛾(𝑡2−𝑡1)} via the mean value theorem, and estimated the last two integrals exactly as the integral in (48). We note that 𝐶₃ and 𝐶₄ have the same dependence as 𝐶₂, which is independent of 𝑞 since 𝐶₁ is independent of 𝑞.

It therefore follows from (48) and (49) that ℱ maps 𝐶([0,𝑇]) into a convex subset of itself. All that remains for an application of Schauder's fixed point theorem is to show that ℱ is continuous in the norm ‖⋅‖_∞ of the separable Banach space ℬ(𝐶([0,𝑇]),‖⋅‖_∞).

To show that (ℱ𝑞)(𝑡) is continuous in ‖⋅‖_∞, we consider ℱ𝑞₁(𝑡)−ℱ𝑞₂(𝑡)=𝛽_Ω^𝑡₀𝑣𝜙,𝜃,𝜏;𝑞₁−𝑣𝜙,𝜃,𝜏;𝑞₂𝑒^{−𝛾(𝑡−𝜏)}d𝜏dΩ, (50) where 𝑣(𝜙,𝜃,𝜏;𝑞_𝑖) is the weak solution of (20) for continuous 𝑞_𝑖(𝑡), 0≤𝑡≤𝑇, for 𝑖=1,2. Forming the difference of (20) for 𝑞_𝑖(𝑡), for 𝑖=1,2, we have −_𝑄𝑇𝑢₁−𝑢₂𝑔_𝑡sin𝜙d𝑄_𝑇+_𝑄𝑇𝑢_1𝜙−𝑢_2𝜙𝑔_𝜙sin𝜙d𝑄_𝑇+_𝑄𝑇𝑢_1𝜃−𝑢_2𝜃𝑔_𝜃csc𝜙d𝑄_𝑇=−_𝑄𝑇𝛼𝑞₁𝑢₁−𝛼𝑞₂𝑢₂𝑔sin𝜙d𝑄_𝑇+𝛾_𝑄𝑇𝑤₁−𝑤₂𝑔sin𝜙d𝑄_𝑇,∀𝑔∈𝑊^1,1,−_𝑄𝑇𝑣₁−𝑣₂𝑔_𝑡sin𝜙d𝑄_𝑇+_𝑄𝑇𝑣_1𝜙−𝑣_2𝜙𝑔_𝜙sin𝜙d𝑄_𝑇+_𝑄𝑇𝑣_1𝜃−𝑣_2𝜃𝑔_𝜃csc𝜙d𝑄_𝑇=_𝑄𝑇𝛼𝑞₁𝑢₁−𝛼𝑞₂𝑢₂𝑔sin𝜙d𝑄_𝑇−𝛽_𝑄𝑇𝑣₁−𝑣₂𝑔sin𝜙d𝑄_𝑇,∀𝑔∈𝑊^1,1,𝑤₁𝜙,𝜃,𝑡;𝑞₁−𝑤₂𝜙,𝜃,𝑡;𝑞₂=𝛽^𝑡₀𝑣₁𝜙,𝜃,𝑡;𝑞₁−𝑣₂𝜙,𝜃,𝑡;𝑞₂𝑒^{−𝛾(𝑡−𝜏)}d𝜏, (51)(52)(53) where the 𝑢_𝑖, 𝑣_𝑖, and 𝑤_𝑖 depend upon the 𝑞_𝑖, for 𝑖=1,2, as indicated by the 𝑤_𝑖 and 𝑣_𝑖 expressed in (52). Also, in (51) we can write _𝑄𝑇𝛼𝑞₁𝑢₁−𝛼𝑞₂𝑢₂𝑔sin𝜙d𝑄_𝑇=_𝑄𝑇𝛼𝑞₁−𝛼𝑞₂𝑢₁𝑔sin𝜙d𝑄_𝑇+_𝑄𝑇𝛼𝑞₂𝑢₁−𝑢₂𝑔sin𝜙d𝑄_𝑇. (54) Hence, if the 𝑢, 𝑣, 𝑤, and 𝛼(𝑞) as in (20) were, respectively, replaced by 𝑢₁−𝑢₂, 𝑣₁−𝑣₂, 𝑤₁−𝑤₂, and 𝛼(𝑞₂) above, and if the term ∫_𝐷0𝑔𝑓sin𝜙d𝐷₀ was deleted, then the right-hand side of (52) would only contain the term _𝑄𝑇𝛼𝑞₁−𝛼𝑞₂𝑢₁𝑔sin𝜙d𝑄_𝑇 (55) to be estimated, instead of the initial data term ∫_𝐷0𝑔𝑓sin𝜙d𝐷₀. Now the term (55) can be estimated as follows: _𝑄𝑇𝛼𝑞₁−𝛼𝑞₂𝑢₁𝑔sin𝜙d𝑄_𝑇≤_𝑄𝑇𝛼(𝑞₁(𝜏))−𝛼(𝑞₂(𝜏))²𝑢²₁sin𝜙d𝑄_𝑇^1/2×_𝑄𝑇𝑔²sin𝜙d𝑄_𝑇^1/2≤12_𝑄𝑇𝛼(𝑞₁(𝜏))−𝛼𝑞₂(𝜏)²𝑢²₁sin𝜙d𝑄_𝑇+12_𝑄𝑇𝑔²sin𝜙d𝑄_𝑇≤12‖𝛼‖_∞+‖𝛼‖_∞𝑑𝛼𝑑𝑞_∞𝑞₁−𝑞_2∞×_𝑄𝑇𝑢²₁sin𝜙d𝑄_𝑇+12_𝑄𝑇𝑔²sin𝜙d𝑄_𝑇≤𝐶₅𝑞₁−𝑞_2∞+12_𝑄𝑇𝑔²sin𝜙d𝑄_𝑇, (56) where 𝐶₅ depends on the bound for ‖𝑢₁‖_𝑄𝑇 which is given by (39) for the weak solution of (20) and is independent of 𝑞₁. The second term on the right-hand side of (54) and (56) involving the 𝑔 becomes 𝑢₁−𝑢₂ in the energy inequality associated with (52). We now have a replay of (37), (38), and (39), with ‖𝑓‖²_𝐷0 replaced with 𝐶₅‖𝑞₁−𝑞₂‖_∞. From the estimate of ‖𝑣₁−𝑣₂‖²_𝑄𝑇 resulting from the Galerkin estimate for ‖𝑢₁−𝑢₂‖²_𝑄𝑇+‖𝑣₁−𝑣₂‖²_𝑄𝑇+‖𝑤₁−𝑤₂‖²_𝑄𝑇 in terms of 𝐶₆‖𝑞₁−𝑞₂‖_∞, we obtain the estimation of (50) below: ℱ(𝑞₁)−ℱ(𝑞₂)_∞≤𝐶₇𝑞₁−𝑞_2∞. (57) This leads to the following statement.

Theorem 1. From (57), one infers the continuity of the mapping ℱ and the validity of the application of the Schauder fixed point theorem, yielding the existence of a weak solution of the weak formulation of (7), (8), and (9), given by (16), (18), and (19).

Proof. See the preceding analysis.

5. Future Considerations

We leave open for consideration the establishment of a unique solution to the weak formulation developed in this paper. This will entail a numerous amount of estimates, which are too tedious to be included in the present work. We refer the interested reader to Chapter 3, Section 2, of [18], as we anticipate that analogues of the energy inequalities derived therein can be generalized to ones that apply to our current problem. We also anticipate that these inequalities will ultimately produce a contraction mapping via a bootstrapping procedure. The constants in these inequalities will have to be estimated closely enough to show the dependence on (small) 𝑡, in order to permit the construction of the aforementioned contraction mapping.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the referee for the timely insights, suggestions, and inquiries. In addition, they appreciate the assistance of Frank B. Jones (Rice University) for sharing with them his unpublished lecture notes pertaining to the heat equation on the unit sphere.