Existence of solutions for a model of microwave heating

This paper is concerned with a system of differential equations related to a circuit model for microwave heating, complemented by suitable initial and boundary conditions. A RCL circuit with a thermistor is representing the microwave heating process with temperature-induced modulations on the electric field. The unknowns of the PDE system are the absolute temperature in the body, the voltage across the capacitor and the electrostatic potential. Using techniques based on monotonicity arguments and sharp estimates, we can prove the existence of a weak solution to the initial-boundary value problem.


Introduction
In this work, we deal with a problem which arises from a circuit model for microwave heating: in particular, we aim at proving the existence of a solution to a coupled system of three differential equations (an ODE, an elliptic equation and a nonlinear parabolic PDE) and appropriate initial and boundary conditions. More specifically, we consider a traditional RLC circuit in which a thermistor has been inserted: this one has a cylindrical shape and takes into account the temperature's effect (for further details see [4]). The system of equations we focus on is obtained in [4]: it involves the absolute temperature ϑ, the voltage V across the capacitor and the potential φ.
Firstofall, let us introduce the notation that we will use in the following: B ⊆ R 2 smooth bounded domain , Ω = B × (0, ℓ) , (1.1) with ℓ > 0 being the height of the cylinder; so that Γ ℓ denotes the lateral boundary and Γ b collects the basis and top boundaries of the cylinder; where T > 0 denotes the final time. Hence, the thermistor is represented by the cylinder Ω in R 3 , while Q represents the spatiotemporal domain.
Let us point out that throughout the paper we will use the notation x for the variable in the two-dimensional domain B, while z will denote the third variable ranging in (0, ℓ). It is also important to emphasize that the term ∂φ ∂z (t) dx , (1.11) which appears in equation (1.4), in principle could depend on both z and t: actually, we will check that I R only depends on t, i.e., so that I R = I R (t) and the ordinary differential equation (1.4) makes sense.
The system (1.4)-(1.10) turns out to be interesting from a physical point of view. In fact, V represents the voltage across the capacitor and equation (1.4) describes how this one is linked with Ω σ(ϑ) ∂φ ∂z dx, which renders the current across the thermistor. Equation (1.5) is supplied with the mixed boundary conditions of Dirichlet type at Γ b (indeed, (1.8) involves the value of V as well on the upper face) and the no-flux condition (1.9) across the boundary Γ ℓ , for t varying in (0, T ). The third equation (1.6) illustrates the evolution of the temperature with respect to the source of ohmic heating σ(ϑ) |∇φ| 2 . Moreover, the condition (1.10) is specifying the heat flux across the thermistor and, in particular, states the proportionality with the difference of the values of the function h on the inside and outside temperatures, at the boundary Σ ℓ .
Beyond the relevance from the physical point of view, the problem (1.4)-(1.10) is intriguing for a mathematical approach: as a matter of fact, it consists of an ordinary differential equation in V , an elliptic equation in φ and a nonlinear parabolic partial differential equation in ϑ. Nonlinear terms are present in all the three equations and also in the boundary condition (1.10). In this work, the terms we will accurately deal with and that are more difficult to handle are the conductivity σ(ϑ) in the second equation (1.5) and the right hand side σ(ϑ) |∇φ| 2 in (1.6): actually, the former makes a strong coupling for the potential φ and the latter would not be found to have so much regularity.
We mention now about some related literature for microwave heating. In the contribution [9] the time-harmonic Maxwell equations with temperature dependent coefficients in a domain occupied by a conducting medium are coupled with the heat equation controlling the temperature distribution induced by the electric field: this is discussed as a model for microwave heating and the existence of a global solution of the coupled nonlinear system is shown. The paper [5] deals with a mathematical model coupling Maxwell's equations to the enthalpy formulation of the Stefan problem: under suitable conditions on the material properties, the model is proved to admit a global weak solution. An optimization problem for a microwave/induction heating process is studied in [8]: the control variable is the applied electric field on the boundary and the cost function is shaped in order the temperature profile at the final time has a relative uniform distribution in the field. A reduced model for the microwave heating of a thin ceramic slab is investigated in [1] and steady-state solutions and their linear stability properties are discussed.
In conclusion, we briefly outline the contents of our paper.
In Section 2 we arrive at the statement of the noteworthy theorem, which ensures that the problem (1.4)-(1.10) has a solution: in particular, we will write some general variational formulations for the equations and we will state the precise existence result.
In Section 3, we will focus on the proof of the theorem: the idea is to use a delay argument in equation (1.5) and consider a truncation of the term σ(ϑ) |∇φ| 2 in (1.6). More precisely, given τ > 0, we will will prove the existence of a solution for the approximating problem in each interval [nτ, (n + 1)τ ] by using the result contained in [6] and a fixed point argument at each iteration. A solution for the approximating problem will then be obtained "pasting" together accurately all the solutions obtained in every single interval. Section 4 contains the uniform estimates on the solution of the approximating problem that are helpful for the limit procedure. The idea here will be to test equation (1.6) by suitable functions we will later introduce.
Finally, Section 5 collects the arguments we use to pass to the limit in the approximating problem in order to recover a solution for the original one: the main tools here are some compactness results.

The main result
In this section, we present the main existence theorem that will be proved in the paper. Firstly, we introduce a general reformulation of problem (1.4)-(1.10).
We multiply (1.5) by a test function w ∈ H 1 0,Γ b (Ω), with then, taking into account (1.9) and integrating by parts, we easily obtain for t ∈ [0, T ]. Now, in order to eliminate the presence of V (t) in the boundary condition (1.8), let us change the variable φ by introducing in this way, an easy computation shows that conditions (1.8)-(1.9) become since the third component of n is null on the lateral boundary Γ ℓ . Hence, the equality (2.2) can be rewritten as (2.5) Note that, for a fixed t ∈ [0, T ], if σ(ϑ(t)) lies in L ∞ (Ω) and is bounded from below by a positive constant, then definition (2.1) and the Poincaré inequality allow us to conclude that w → Ω σ(ϑ(t))|∇w| 2 dxdz yields an equivalent norm in H 1 0,Γ b (Ω). Within this framework, it is not difficult to check that the Lax-Milgram lemma implies the existence of a unique ψ(t) solving the variational equality (2.5).
It is now time to set some assumptions. About the functions σ and k, we assume that for some positive constants σ * , σ * , k * , k * . Note that, although these functions work in principle only on (0, +∞) due to the physical meaning of the variable ϑ, in case they can be easily extended with reasonable values to the whole of R. About the initial data, we let V 0 , V ′ 0 ∈ R , ϑ 0 ∈ L 2 (Ω) (2.10) so that ψ 0 ∈ H 1 0,Γ b (Ω) and φ 0 ∈ H 1 (Ω) are well defined. where we have taken c 0 = 1 to simplify the notation. A variational formulation of (2.14) can be easily obtained by multiplying it by a test function w and integrating by parts on account of the boundary condition (1.10). In particular, if we introduce the auxiliary variable u = K(ϑ) , (2.15) let β(r) = h(γ(r)), r ∈ R, (2.16) and set we arrive at the following formulation in terms of u: (2.18) Please note that, here and in the sequel, we use the notation "dv" (in place of the heavy dxdz) for the volume element in the integrals over Ω and "ds" for the surface element in the integrals on the boundaries of Ω. We also observe that in (2.18) the test functions w are taken in the smoother space W 1,∞ (Ω) in order to try to give a meaning to all the integrals, in particular to the first one on the right hand side. Actually, we can set a larger space for the test functions, as it is pointed out in the next statement.
Hence, we are ready to present the main result of the paper, which ensures that the problem we are dealing with actually has a solution (V, φ, ϑ): moreover, it specifies which spaces are considered and in which sense the solution is intended. hold. Moreover, there are two constants ϑ * > 0 and C β > 0 such that where β is the (convex and C 1 ) primitive of β such that β(0) = 0. Then, there exists a quintuplet (V, φ, ψ, ϑ, u) such that for some conjugate exponents p ∈ (1, 5/4) and q ∈ (5, +∞), 1 p + 1 q = 1, and satisfying We observe that the initial condition for ϑ in (2.30) makes sense at least in W 1,q (Ω) ′ due to (2.24), (2.10) and the inclusion L 2 (Ω) ⊂ W 1,q (Ω) ′ . Moreover, as q > 3, let us point out that, owing to the Sobolev embedding results, not only Remark 2.3. Please note that condition (2.22) is a priori very restrictive as a growth condition for β. However, hypothesis (2.22) can be considered acceptable if we bear in mind our application: the input of the function β is u, i.e. the temperature (modified through the Lipshitz-operator K), so that (2.22) trivially holds true if we confine our system to the case in which the temperature assumes values in a bounded interval. This is cleary acceptable from the physical interpretation: for example, we can deal with all situations in which the temperature does not exceed an arbitrary high level. Thus, this apparently restrictive mathematical hypothesis does not affect the interpretation of the problem.
Remark 2.4. Assumption (2.21) establishes a positive bound from below for the initial and some boundary values of the absolute temperature, which is completely reasonable and will help us to infer the same bound for the variable ϑ. About condition (2.22), we would like to comment also on the right hand side of the inequality: by requiring that the growth of β is controlled by a linearly growing function, we can find some useful estimate for suitable norms of ∂ϑ ∂t in order to pass to the limit; furthermore, the application of the result of [6] during the delay argument needs that |β| is somehow controlled by β plus a constant.

Approximating the problem
As we have anticipated, the idea is to prove Theorem 2.1 using a delay argument, that we will describe in this section. Firstly, let us introduce the delay parameter τ ∈ (0, T ) and define the value of ϑ also for negative times in the following way: Let now focus on the interval [0, τ ] and consider the system (2.25)-(2.27), in which we introduce some delay terms involving ϑ(t−τ ) and a truncation in |∇φ(t)| 2 : more precisely, we use ) and the right hand side of (2.27) , where T τ is the truncation operator In other words, we consider the approximated system for almost every t ∈ [0, τ ] and Thus, we look for a solution on the interval [0, τ ].
The idea at this level is to use a fixed point argument: let us explain in a first intuitive approach how we will proceed. We fix a suitable V in equation (3.4), which becomes in this way an explicit elliptic equation; we solve it and recover the solution ψ. From (3.6), with V in place of V , we are able to find the corresponding φ and substitute it in equation (3.3): thus, this one becomes a linear ordinary differential equation with known terms on the right hand side. Then, if we consider the Cauchy problem given by (3.3) and the initial condition in (3.8), we obtain a unique solution V in [0, τ ]. Now, by considering the application mapping V into V , we will deduce some contraction estimates provided τ is sufficiently small. Consequently, if the interval [0, τ ] is suitably chosen, then there is a unique fixed point V that solves the system involving (3.3), (3.4). Of course, the corresponding ψ and φ can also be determined easily from (3.4) and (3.6). Once we have found the triplet (V, φ, ψ), we can check that the result contained in [6] applies to equation (3.5), complemented by the relation (3.7) and the initial condition in (3.8); moreover, since the terms on the right hand side of (3.5) are smoother, in particular σ(ϑ( · − τ ))T τ |∇φ| 2 is uniformly bounded, there will be a solution pair (ϑ, u) with ϑ continuous from [0, τ ] to L 2 (Ω).
At this point, the idea is to focus on the interval [τ, 2τ ] and repeat the same argument used in [0, τ ] for the new interval, with the new initial values V (τ ), ϑ(τ ) and with ϑ( · −τ ) specified by the solution component ϑ found at the previous step. Note that in this case (3.3) should be rewritten as for t ∈ [τ, 2τ ], and the term −λ 3 τ 0 V (r) dr on the right hand side is now a datum since the solution V has been already found from the previous iteration. Then, one should continue in this way for every interval [nτ, (n + 1)τ ], with n going from 2 to a value N with (N + 1)τ ≥ T . Hence, if it is possible to correctly paste together the solutions at each interval, then we will find a solution of the approximated problem in the whole interval [0, T ]. The most important thing is that we have to check that the contraction estimates used in the fixed point argument at each iteration do not depend on the specific interval we consider, or, in other words, that there exists some τ > 0, small enough and independent of the sub-intervals, such that the contraction estimates hold.
Let us now present the proof of the existence of a solution for the approximating problem. The key of the argument is the following lemma, which shows that some contraction estimates hold for a suitable value of τ on each sub-interval.
and solving the problem for all t ∈ [ρ, ρ + τ ], and and let V 1 , V 2 denote the corresponding solution components of the problem (3.9)-(3.13). Then, the following estimate holds: 14) where |B| denotes the bidimensional measure of the set B in (1.1).
Proof. For all t ∈ [ρ, ρ + τ ], it natural to introduce the bilinear form a t : Using (2.9) and the Poincaré inequality, it is easy to check that a t is continuous and coercive on H 1 0,Γ b (Ω) . Moreover, we see that the linear functional Moreover, testing the previous expression by w = ψ(t) ∈ H 1 0,Γ b (Ω) and using the Young inequality, it easily follows that (3.12). At this point, we note that the same argument leading to (2.6) ensures that the term does not depend on z, and equation (3.10) makes sense. Then, equation (3.10) becomes an ordinary differential equation in [ρ, ρ + τ ] with appropriate initial conditions (3.13). By virtue of (2.9)-(2.11), (2.19) and V ∈ C 0 [0, ρ] , the right hand side of (3.10) is a continuous function in [ρ, ρ + τ ]. Hence, it is straightforward to see that there exists a unique solution of the Cauchy problem expressed by (3.10) and (3.13). Thus, we have proved the first part of the lemma. Let us focus now on the contraction estimates. Given be the corresponding solutions of problem (3.9)-(3.13). Then, if we consider (3.10) and take the difference, testing by Hence, integrating on (ρ, t) we deduce that and the Hölder inequality and condition (2.9) allow us to infer . Now, taking (3.12) and (3.16) into account, for all t ∈ [ρ, ρ + τ ] we have At this point, passing to the square roots it is straightforward to derive the final estimate (3.14).
Another auxiliary result is formulated in the next lemma.
We are now ready to describe how to obtain a solution of the approximating problem. The idea is to build a solution (V τ , φ τ , ψ τ , ϑ τ , u τ ) step by step.
Let us define ϑ τ (t) for t ≤ 0 as in (3.1) and focus on the first interval [0, τ ]; consider (Ω) which solves the problem (3.10)-(3.13). Hence, it is natural to introduce the operator and observe that, thanks to (3.14), we can fix τ such that that is, in order that (3.21) be a contraction mapping. Hence, there exists a unique which is a fixed point for Λ 0 , i.e. a solution, along with the related Next, we choose ρ = 0, ϑ(t) = ϑ τ (t − τ ) = ϑ 0 and φ(t) = φ (0) (t) for t ∈ [0, τ ], besides the initial value Θ 0 = ϑ 0 , in Lemma 3.2 and find a pair (ϑ, u) = (ϑ (0) , u (0) ) with Let us construct now the solution of approximating problem on the sub-interval [τ, 2τ ]. Here, the delay terms are associated to the functions found in the previous passage. We consider thus the problem The same argument that we have used to prove the existence of a solution on the first interval can be easily repeated in this case in the same way. Take Find the corresponding solution (V, φ, ψ) and observe that the obviously defined operator is a contraction mapping as well, due to (3.22) and (3.14): this follows from the fact that the constant present in estimate (3.14) does not depend on ρ. Consequently, we are able to find a solution of the problem (3.23), (3.24), (3.26), and V (1) (τ ) = V τ (τ ). Then, we apply Lemma 3.2 with ϑ(t) = ϑ τ (t − τ ) and φ(t) = φ (1) (t) for t ∈ [τ, 2τ ], choosing the initial value Θ τ = ϑ τ (τ ). Thus, we obtain a solution of the problem (3.25), (3.27), and the second initial condition in (3.28). At this point, we can extend our solution to the interval [τ, 2τ ] in the following way: Note that, by this definition, the pasted functions ϑ τ and u τ remain continuous from [0, 2τ ] to L 2 (Ω), and consequently also φ τ , ψ τ keep the continuity property in [0, 2τ ]; then, by arguing on (3.23) and (3.28), we realize that V τ ∈ C 1 [0, 2τ ] .
To reach exactly the final time T , it suffices to take τ = T /(n + 1) with n sufficiently large in order to satisfy (3.22); otherwise, we solve the last iteration in the shorter interval [nτ, T ] with T < (n + 1)τ . It is now straightforward to deduce that (V τ , φ τ , ψ τ , ϑ τ , u τ ) is a solution for the approximating problem (3.3)-(3.8).
Let us collect the result we have just proved in the following statement.
and satisfying

Uniform estimates
In this section, we present some estimates, independent of τ , on the solution of the approximating problem (3.31)-(3.36) specified by Theorem 3.3. Then, we aim to pass to the limit and recover a solution to the original problem (2.25)-(2.30).

The second estimate
The aim of this subsection is to prove the existence of some positive constants C 1 , C 2 , independent of τ , such that the following estimates hold: We multiply equation (3.31) by V τ (t) and integrate. Using (2.9), the independence of z of the integrals over B (cf. (2.6)) and the Young inequality, we deduce that for all ε > 0. If we make the choice ε = λ 2 ℓ ℓ + σ * |Ω| , we infer that for all t ∈ [0, T ]. At this point, if we take w = ψ τ (t) ∈ H 1 0,Γ b (Ω) in (3.32), arguing as for (3.16) we obtain Therefore, with the help of (3.34) and (2.9) we easily deduce that Now, if we substitute (4.6) in (4.4) and recall (2.11), we have that for all t ∈ [0, T ]. Hence, we can apply the Gronwall lemma and conclude that {V τ } is bounded in L ∞ (0, T ) independently of τ . Moreover, taking this into account in (4.5)-(4.6) and recalling that σ(r) ≥ σ * > 0 for all r ∈ R, thanks to the Poincaré inequality and (3.34) we obtain the estimate (4.3). Finally, using condition (4.3) and the bound for V τ L ∞ (0,T ) , by a comparison of terms in (3.31) we deduce (4.2).

The fourth estimate
This is the really important estimate we are going to prove. For convenience, let us ask the reader to state the estimate, unusually, at the end of the subsection. We have been inspired by a computation already performed in [3, pp. 1361-1362].
We proceed by taking w ∈ W 1,q (Ω) in equation (3.33): isolating the first term, thanks to the Hölder inequality and conditions (2.9) and (2.22), we obtain Recall that for all r ∈ [1, +∞) there exists a constant M r > 0 such that (see, e.g., [2]) and that the following embedding W 1,q (Ω) ⊆ L ∞ (Ω) is continuous since q > 3 and Ω ⊆ R 3 ; hence, using (4.3) and (4.19), from the previous calculations we deduce that for all w ∈ W 1,q (Ω) and for some constants D 7 , D 8 independent of τ . Then, we have that for almost all t ∈ [0, T ]; taking the pth power of this expression and integrating on (0, T ), as p < 2 and (4.19) holds we infer and (4.21) follows easily from the above inequality.

The passage to the limit
In this last section, we collect all the arguments needed to pass to the limit in the approximating problem (3.31)-(3.36) as τ ց 0 and recover in this way a solution of the original problem (2.25)-(2.30).
We are now ready to prove (5.15). Indeed, condition (5.17) implies that there is a further subsequence of {τ n } n∈N , for which we keep the same notation, such that ϑ τn (· − τ n ) → ϑ a.e. in Q ; on account of (2.9), the continuity of σ implies that σ ϑ τn (· − τ n ) → σ ϑ a.e. in Q , as n → ∞, (5.18) and this convergence and the boundedness of σ ensure that we can apply the dominated convergence theorem and thus deduce (5.15).

Conclusion of the proof
By virtue of (5.21) and (5.15), now we can take the limit in the nonlinear term of equation (3.31) and obtain (2.25), using (5.11) to pass to the limit in the left hand side. To deal with (3.33), we need to show that σ ϑ τn (· − τ n ) T τn |∇φ τn | 2 → σ ϑ |∇φ| 2 in L 1 (Q) as n → ∞ . To this aim, we take advantage of the property stated in the next lemma.
Lemma 5.4. Let T τ the truncation operator defined in (3.2) and let {z n } n∈N ⊆ L 1 (Q) such that z n → z in L 1 (Q). Then, it turns out that T τn (z n ) → z in L 1 (Q) as n → ∞.