1 Introduction

We refer to the semilinear equation

$$ \mathcal{L}_ \varepsilon u= F(x,t,u) $$
(1)

where \(\mathcal{L}_{\varepsilon}\) is the third-order parabolic operator:

$$ \mathcal{L}_ \varepsilon=(\partial_{xx}-\lambda \partial_x) (\varepsilon \partial_t+1 ) - \partial_t(\partial_t+\alpha). $$
(2)

Equation (1) characterizes the evolution of several dissipative models such as the motions of viscoelastic fluids or solids [1–4]; the sound propagation in viscous gases [5]; the heat conduction at low temperature [6, 7] and the propagation of localized magnetohydrodinamic models in plasma physics [8]. Moreover, it can also be referred to reaction diffusion systems [9].

As example of perturbed model of the phase evolution, we will consider the non linear phenomenon concerning the Josephson effects in superconductivity.

More precisely, if φ=φ(x,t) is the phase difference in a rectangular junction and γ is the normalized current bias; when λ=0 and F=sinφ−γ, Eq. (1) gives the well-known perturbed Sine-Gordon equation (PSGE) [10]:

$$ \varepsilon\varphi_{xxt}+\varphi_{xx}- \varphi_{tt}-\alpha\varphi_{t}=\sin \varphi-\gamma. $$
(3)

The terms εφ xxt and αφ t characterize the dissipative normal electron current flow respectively along and across the junction. They represent the perturbations with respect to the classic Sine Gordon equation [10, 11]. When the surface resistance is negligible, then ε(<1) is vanishing and a singular perturbation problem for Eq. (3) could appear [12]. As for the coefficient α of (3), it depends on the shunt conductance [13] and generally one has a<1 [14–16]. However, if the resistance of the junction is so small as to short completely the capacitance, the case a>1 arises [10, 17].

More recently, the case of the exponentially shaped Josephson junction (ESJJ) has been considered. The evolution of the phase inside this junction is described by the third order equation:

$$ \varepsilon\varphi_{xxt}+\varphi_{xx}- \varphi_{tt} - \varepsilon \lambda\varphi_{xt} -\lambda \varphi_x- a\varphi_{t}=\sin \varphi-\gamma $$
(4)

where λ is a positive constant generally less than one [18, 19] while the terms λφ x and λεφ xt represent the current due to the tapering. In particular λφ x correspond to a geometrical force driving the fluxons from the wide edge to the narrow edge [19, 20].

According to recent literature [14, 18, 20–23], an exponentially shaped Josephson junction provides several advantages with respect to a rectangular junction. For instance in [18] it has been proved that in an ESJJ it is possible to obtain a voltage which is not chaotic anymore, but rather periodic excluding, in this way, some among the possible causes of large spectral width. It is also proved that the problem of trapped flux can be avoided. Moreover, some devices as SQUIDs were built with exponentially tapered loop areas [24].

The analysis of many initial—boundary problems related to the PSGE (3) has been discussed in a lot of papers. In particular in[25], to deduce an exhaustive asymptotic analysis, the Green function of the linear operator

$$ \mathcal{L}=\partial_{xx}\bigl(\varepsilon \partial_t+c^2 \bigr) - \partial_t( \partial_t+\alpha) $$
(5)

has been determined by Fourier series. By means of its properties an exponential decrease of both linear and non linear solutions is deduced.

The aim of this paper is the analysis of the Dirichlet boundary value problem related to Eq. (4).

The Green function G of the linear strip problem is determined by Fourier series and properties of rapid convergence are established. So, when the source term F is a linear function, then the explicit solution is obtained and an exponential decrease of the solution is deduced.

When F is nonlinear, the problem is reduced to an integral equation with kernel G and an appropriate analysis implies results on the existence and uniqueness of the solution. Moreover, by means of suitable properties of G, a priori estimates, continuous dependence upon the data and asymptotic behaviour of the solution are achieved, too.

2 Statement of the Problem and Properties of the Green Function

Let l,T be arbitrary positive constants and let

$$\varOmega_T =\bigl\{(x,t) : 0 < x < l, 0 < t \leq T \bigr\}. $$

The boundary value problem related to equation (1) is the following:

$$ \left \{ \begin{array}{l} (\partial_{xx} - \lambda \partial_x ) (\varepsilon u_{t}+ u) - \partial_t(u_{t}+\alpha u) =F(x,t,u),\quad (x,t)\in\varOmega_T,\\ u(x,0)=h_0(x),\qquad u_t(x,0)=h_1(x),\quad x\in[0,l],\\ u(0,t)=0, \qquad u(l,t)=0, \quad 0<t \leq T. \end{array} \right . $$
(6)

By Fourier method it is possible to determine the Green function of the linear operator \(\mathcal{L}_{\varepsilon}\). So, let

$$ \gamma_n=\frac{n\pi}{l}, \qquad b_n= \bigl(\gamma_n^2 + \lambda^2/4 \bigr), \qquad g_n=\frac{1}{2} (\alpha + \varepsilon b_n ), \qquad \omega_n=\sqrt{g_n^2- b_n } $$
(7)

and

$$ G_n(t)= \frac{1}{\omega_n} e^{-g_n t} \operatorname{sinh}(\omega_nt), $$
(8)

by standard techniques, the Green function can be given the form:

$$ G(x,t,\xi)=\frac{2}{l} e^{\frac{\lambda }{2 } x} \sum _{n=1}^{\infty} G_n(t) \sin \gamma_n\xi \sin\gamma_nx. $$
(9)

This series is endowed of rapid convergence and it is exponentially vanishing as t tends to infinity. In fact, if we denote by

$$ a_\lambda= \alpha + \varepsilon \lambda^2/4 $$
(10)

and

$$ p_\lambda= \frac{\pi^2}{\varepsilon \pi^2 + a_\lambda l^2}, \qquad q_\lambda= \frac{ a_\lambda + \varepsilon(\pi/l)^2}{2}, \qquad \delta \equiv \min(p_\lambda, q_\lambda), $$
(11)

the following theorem holds:

Theorem 1

Whatever the constants α,ε,λ may be in ℜ+, the function G(x,ξ,t) defined in (9) and all its time derivatives are continuous functions in Ω T and it results:

$$ \bigl \vert G(x,\xi,t)\bigr \vert \leq M e^{-\delta t}, \qquad \biggl \vert \frac{\partial^j G}{\partial t^j}\biggr \vert \leq N_j e^{-\delta t},\quad j \in \mathbb{N} $$
(12)

where M,N j are constants depending on α,λ,ε.

Proof

Physical problems lead to consider αε<1 and denoting by

$$ N_{1,2}^{\lambda}= \frac{l}{2 \varepsilon^2\pi} \bigl[ 4 (1\mp \sqrt{1-\alpha \varepsilon} )^2 - {\varepsilon^2 \lambda^2} \bigr]^{1/2}, $$
(13)

let us assume that \(N_{1,2}^{\lambda} >1 \). So, let k be a positive constant less than one and let \(\bar{N}_{1,2} , N_{k}^{\lambda}\) be the lowest integers such that

$$ \left \{ \begin{array}{l} \bar{N}_1 < N_1^{\lambda}, \qquad\bar{N}_2 > N_2^{\lambda}; \\[6pt] N_k^{\lambda} > \dfrac{l}{2 \varepsilon^2\pi {k}} [ 4 (1\mp \sqrt{1-\alpha k \varepsilon} )^2 - {\varepsilon^2 k \lambda^2} ]^{1/2}. \end{array} \right . $$
(14)

We start analysing the hyperbolic terms when \(n\geq\bar{N}_{2}\). Letting

$$ X_n=\frac{b_n}{g_n^2} <1, \qquad \varphi_n=g_n(-1+ \sqrt{1-X_n}), $$
(15)

it is possible to prove that \(\varphi_{n} \leq - \frac {\gamma_{n}^{2}}{2 g_{n}}\). So it results:

$$ e^{-t(g_n-\omega_n)}\leq e^{-p_\lambda t}. $$
(16)

Furthermore, it is easily verified that for all \(n \geq N^{\lambda}_{k} ( \geq\bar{N}_{2}) \) it results \(\frac{b_{n}}{g_{n}^{2}} \leq k \) and hence one has:

$$ \omega_n \geq g_n (1-k)^{1/2} \geq n^2 \frac{2 l^2}{\varepsilon \pi^2} (1-k)^{1/2}. $$
(17)

Other terms can be treated similarly. For instance, as for circular terms, it can be proved that \(e^{- g_{n} t}\leq e^{-q_{\lambda}t}\).

In consequence estimate (12)1 holds ∀n≥1.

As for (12)2, one has:

$$ g_n - \omega_n = \frac{b_n}{g_n + \omega_n } \leq \frac {2}{\varepsilon }+\frac{\lambda^2}{4 \varepsilon q_\lambda }, \quad\forall n \geq1 $$
(18)

and by means of standard computations, (12)2 can be deduced, too.

It may be similarly proved that the theorem holds also when αε≥1 or when the conditions \(N_{1,2}^{\lambda} > 1 \) do not hold.

Finally we notice that when \(N_{1,2}^{\lambda}\) are integers, the constants M and N j in (12) could depend on t. □

As for the x-derivatives of Fourier series like (9), attention is needed towards convergence problems. For this, we will consider x-differentiations of the operator (ε∂ t +1)G instead of G and G t .

Theorem 2

Whatever α,ε,λ may be, the function G(x,ξ,t) defined in (9) is such that:

$$ \bigl \vert \partial_{x}^{(i)} (\varepsilon G_t + G )\bigr \vert \leq A_i e^{-\delta t} \quad(i=0,1,2) $$
(19)

where δ is defined in (11) and A i (i=0,1,2) are constants depending on a,ε,λ.

Proof

As for the hyperbolic terms in G, it results:

$$ \varepsilon G_t+G=e^{\frac{\lambda x }{2 }} \sum _{n=1}^{\infty} \frac{e^{-g_nt}}{l \omega_n} \bigl\{\bigl[1- \varepsilon( g_n-\omega_n)\bigr] e^{\omega_nt}- \bigl[1-\varepsilon(g_n+\omega_n)\bigr]e^{-\omega_nt} \bigr\}. $$
(20)

where according to (15), it results:

$$1-\varepsilon( g_n-\omega_n) = 1 +\varepsilon \varphi_n. $$

So, by means of Taylor’s formula, one has:

$$ 1-\varepsilon ( g_n-\omega_n) = 1- \frac{\varepsilon}{2} g_n X_n - \frac{\varepsilon}{8} g_n X_n^2 -\frac{3}{16} \varepsilon g_n \int_0^{X_n}\frac{(X_n-y)^2}{(1-y)^{5/2}} dy. $$
(21)

Besides, it is possible to prove that ∀n≥1 one has:

$$ X_n < \frac{c^2}{n^2}\quad \bigl(c = l \sqrt{4+ \lambda^2}/ \varepsilon \pi\bigr) $$
(22)

and for all n≥c(1+c 1)(c 1>0) it results:

$$ \int_0^{X_n}\frac{(X_n-y)^2}{(1-y)^{5/2}} dy \leq\frac{2}{3} X_n^2 \biggl[ \frac{(c_1+1)^3}{c_1(c_1+2)]^{3/2}}-1 \biggr]. $$
(23)

So, taking into account that

there exists a positive constant k 1 such that:

$$ |1+\varepsilon \varphi_n | \leq\frac{1}{n^2} \biggl( \frac{\alpha l^2}{\varepsilon\pi^2} +\frac{k_1}{n^2} \biggr). $$
(24)

Estimates of Theorem 1 together with (24) show that the series terms related to the operator εG t +G have order at least of n −4. So it can be differentiated term by term with respect to x and the estimate (19) can be deduced. □

As solution of the equation \(\mathcal{L}_{\varepsilon}v = 0\) we will mean a continuous function v(x,t) which has continuous the derivatives v t ,v tt ,∂ x (εv t +v),∂ xx (εv t +v) and these derivatives verify the equation.

So, we are able to prove the following theorem:

Theorem 3

The function G(x,t) defined in (9) is a solution of the equation

$$ \mathcal{L}_\varepsilon G =(\partial_{xx} - \lambda \partial_x) (\varepsilon G_{t}+ G) - \partial_t(G_{t}+ \alpha G)=0. $$
(25)

Proof

The uniform convergence proved in Theorems 1–2 allows to deduce that:

$$ (\partial_t +\alpha)\frac{\partial G}{\partial t}= \frac{2 e^{\frac{\lambda x }{2 }}}{l} \sum_{n=1}^{\infty } \bigl\{ \bigl[ b_n (\varepsilon g_n -1 )G_n -\varepsilon b_n e^{-g_nt}\cosh \omega_nt \bigr]\sin\gamma_n\xi \sin \gamma_nx \bigr\}, $$
(26)
(27)

Moreover, being

(28)

(25) can be deduced. □

3 Properties of the Convolution

To achieve the solution of the strip problem (6), the convolution of the function G with the data must be analysed. For this, let h(x) be a continuous function on (0,l) and let:

(29)
(30)

The following theorems hold:

Theorem 4

If the data h(x) is a C 1(0,l) function, then u h defined by (29) is a solution of the equation \(\mathcal{L}_{\varepsilon}=0\) and it results:

$$ \lim_{t \rightarrow0}u_h(x,t)=0,\qquad \lim_{t \rightarrow0} \partial_t u_h(x,t)=h(x), $$
(31)

uniformly for all x∈[0,l].

Proof

The absolute convergence of u h with its partial derivatives is proved by means of Theorems 1 and 2 and continuity of function h(x). So, since (25) \(\mathcal{L}_{\varepsilon}u_{h}=0 \) is verified, while Theorem 1 and hypotheses on h(x) imply (31)1.

More being:

$$ \frac{\partial G}{\partial t} =-\frac{2}{\pi} \frac {\partial }{\partial\xi} \sum _{n=1}^{\infty} \frac{\partial G_n}{\partial t} \frac{\cos\gamma_n\xi }{n} \sin \gamma_n x $$
(32)

and

(33)

denoting by η(x) the Heaviside function, it results:

$$ \lim_{t \rightarrow0}\partial_t u_h= \frac{x}{l} \bigl[ h(l)-h(0) \bigr]+h(0)- \int_{0}^{l} h^{\prime}(\xi) \biggl[ \eta(\xi-x)+\frac{x}{l}-1 \biggr] d\xi=h(x). $$
(34)

 □

Theorem 5

Let h(x) be a C 3(0,l) function such that h (i)(0)=h (i)(l)=0(i=1,2,3). Then \(u_{h}^{*}\) defined in (30) is a solution of the equation \(\mathcal{L}_{\varepsilon}=0\) and it results:

$$ \lim_{t \rightarrow0}u_h^*(x,t)=h,\qquad \lim_{t \rightarrow0} \partial_t u_h^*(x,t)=0 $$
(35)

uniformly for all x∈(0,l).

Proof

Properties of h(x) assure that:

(36)

So, since Theorem 3, \(\mathcal{L}_{\varepsilon}u^{*}_{h}=0 \) is verified. Moreover, being:

$$ \partial_t u_h^* = (\partial_{xx}- \lambda\partial_x) u_h $$
(37)

(36) implies (35)2, too. Finally, owing to (31) and (36) , one obtains:

$$ \lim_{t \rightarrow0}u_h^*=\lim_{t \rightarrow0} \biggl[ \partial_t u_h +\varepsilon \biggl( \frac{\lambda^2}{4}u_h - u_{h^{\prime\prime}} \biggr) \biggr] = h(x). $$
(38)

 □

4 Solution of the Linear Problem

Let us consider the homogeneous case. From Theorems 4, 5 the following result is obtained:

Theorem 6

When F=0 and the initial data h 1(x), and h 0(x) verify the hypotheses of Theorems 4–5, then the function:

$$ u(x,t)=u_{h_1}+(\partial_t+\alpha + \varepsilon \lambda\partial_{x} -\varepsilon\partial_{xx}) u_{h_0} $$
(39)

represents a solution of the homogeneous strip problem (6).

Otherwise, when F=f(x,t), let consider

$$ u_f(x,t) = - \int_{0}^{t} d\tau \int_{0}^{l} f(\xi,\tau) G(x,\xi,t-\tau) d \xi. $$
(40)

Standard computations lead to consider at first the problem (6) with g 0=g 1=0. For this the following theorem is proved:

Theorem 7

If the function f(x,t) is a continuous function in Ω T with continuous derivative with respect to x, then the function u f represents a solution of the nonhomogeneous strip problem.

Proof

Since (31)1 it results:

$$ \partial_tu_f(x,t) = \int _{0}^{t} d\tau \int_{0}^{l} f(\xi,\tau) G_t(x,\xi,t-\tau) d\xi $$
(41)

and as proved in Theorem 4, one obtains:

$$ \lim_{\tau\rightarrow t}\partial_tu_f(x,t) = f(x,t). $$
(42)

Hence, one has:

$$ \partial^2_tu_f = f(x,t)+ \int _{0}^{t} d\tau \int_{0}^{l} f(\xi,\tau) G_{tt}(x,\xi,t-\tau) d\xi $$
(43)

and Theorem 3 assures that \(\mathcal{L}_{\varepsilon}u_{f} = f(x,t)\).

Furthermore, owing to (40)–(41) and estimates (12), if B i (i=1,2) are two positive constants, it results:

$$ \vert u_f\vert \leq B_1 \bigl(1-e^{-\delta t}\bigr); \qquad |\partial_tu_f| \leq B_2 \bigl(1-e^{-\delta t}\bigr) $$
(44)

from which initial homogeneous conditions follow. □

The uniqueness is a consequence of the energy-method and we have:

Theorem 8

When the source term f(x,t) satisfies Theorem 7 and the initial data (h 0,h 1) satisfy Theorem 6, then the function

$$ u(x,t)=u_{h_1}+(\partial_t+\alpha + \varepsilon \lambda\partial_{x} -\varepsilon\partial_{xx}) u_{h_0}+u_f $$
(45)

is the unique solution of the linear non-homogeneous strip problem (6).

5 Solution of the Non-linear Problem

Let us consider now the non-linear problem:

$$ \left \{ \begin{array}{l} (\partial_{xx} - \lambda \partial_x ) (\varepsilon u_{t}+ u) - \partial_t(u_{t}+\alpha u) = F(x,t, u),\quad (x,t)\in\varOmega_T,\\ u(x,0)=h_0(x),\qquad u_t(x,0)=h_1(x),\quad x\in[0,l],\vspace{2mm} \\ u(0,t)=0,\qquad u(l,t)=0,\quad 0<t \leq T. \end{array} \right . $$
(46)

As for the data F and h i (x) (i=0,1) we shall admit:

Assumption 9

The functions h i (x) (i=0,1) are continuously differentiable and bounded together with \(h'_{1}(x)\) and \(h^{(k)}_{0}\) (k=1,2). The function F(x,t,u) is defined and continuous on the set

$$ D_T \equiv\bigl\{ (x,t,u) : (x,t) \in \varOmega_T , -\infty < u <\infty \bigr\} $$
(47)

and more it is uniformly Lipschitz continuous in (x,t,u) for each compact subset of Ω T . Besides, F is bounded for bounded u and there exists a constant C F such that the estimate

$$ \bigl \vert F (x,t,u_1) - F (x,t,u_2)\bigr \vert \leq C_F | u_1-u_2 | $$
(48)

holds for all (u 1,u 2).

When the problem (46) admits a solution u then, properties of G and the Assumption 9 ensure that u must satisfy the integral equation

(49)

and it is possible to prove that [26–28]

Theorem 10

The non linear problem (46) admits a unique solution if and only if the integral equation (49) has a unique solution which is continuous on Ω T .

Moreover, let \(\| v \|_{T} = \sup_{\varOmega _{T}} | v (x,t) | \) and let \(\mathcal{B}_{T} \) denote the Banach space

$$ \mathcal{B}_ T \equiv \bigl\{ v ( x,t ) : v \in C ( \varOmega_T ),\ \|v\|_T < \infty \bigr\}. $$
(50)

By means of standard methods related to integral equations it is possible to prove that the mapping ψ defined by (49) is a contraction of \(\mathcal{B}_{T} \) in \(\mathcal{B}_{T}\) and so it admits a unique fixed point u(x,t). In consequence the following theorem holds:

Theorem 11

When the initial data h i (i=0,1) and the source term F verify the Assumption 9, then the problem (46) admits a unique regular solution.

6 Applications

All these results allow us to obtain continuous dependence upon the data, a priori estimates of the solution and asymptotic properties.

According to Assumption 9, let

So, by means of the following theorem the dependence upon the data can be proved:

Theorem 12

Let u 1,u 2 be two solutions of the problem related to the data (h 0,h 1,F 1) and (γ 0,γ 1 F 2) which satisfy the Assumption 9. Then, there exists a positive constant C such that

$$\Vert u_1-u_2 \Vert _T \leq C \sup_{\varOmega_T } \vert h_0-\gamma_0 \vert + C \sup_{\varOmega_T} \vert h_1-\gamma_1 \vert + C \sup_{D_T} \bigl \vert F_1(x,t,u) -F_2(x,t,u) \bigr \vert , $$

where C depends on C F ,T and on the parameters α,ε,λ.

The integral equation and the properties proved for Green Function G imply a priori estimates, too.

Theorem 13

When the data (h 0,h 1,F) of the problem (46) verify the Assumption 9, then the following estimate holds:

$$ \bigl \Vert u(x,t) \bigr \Vert _T \leq \frac{1}{\delta} \bigl(1- e^{-\delta t} \bigr) \Vert F \Vert + K \bigl[ \Vert h_1 \Vert + \Vert h_0 \Vert + \| h_0^{\prime\prime} \| \bigr] e^{- \delta t } $$
(51)

where the constants δ- defined in (11) and K depend on α,ε,λ.

As for the asymptotic properties, obviously the behaviour of the solution depends upon the shape of the source term.

For instance, in the linear case one has:

Theorem 14

When the source term f(x,t) satisfies the condition:

$$ \bigl \vert f(x,t) \bigr \vert \leq C e^{- m t} \quad (C, m = \mathrm{const} > 0), $$
(52)

one has:

$$ \bigl \vert u(x,t) \bigr \vert \leq k e^{- m^* t}, \quad m^* = \min\{\delta, m\},\ k=\mathrm{const}. $$
(53)

An exponentially decreasing behaviour is also possible in the non linear case. In fact, according to [29], let us consider a normed space where

$$ \bigl \Vert u(x,t) \bigr \Vert = \max_{x \in(0,l)} \bigl \vert u(x,t) \bigr \vert $$
(54)

is such that

$$ \bigl \Vert u(x,t) \bigr \Vert \leq \beta e^{- \delta t} $$
(55)

being β a positive constant and δ is defined in (11). Furthermore, let us introduce the following definition [29]:

Definition 15

When the function F is such that |F(x,t,u)|≤γ∥u∥e −δt, then F is an exponential Lipschitz function.

So the following theorem can be proved:

Theorem 16

If the non linear source F is an exponential Lipschitz function, then the solution of the semilinear problem (46) vanishes as follows:

$$ \bigl \vert u(x,t)\bigr \vert \leq K_1 e^{ - \delta t} $$
(56)

where K 1 is a positive constant depending on α,ε,λ.

Since |sinu|≤|u|, a similar behaviour is also verified for the model of superconductivity when F(x,t,u)=sinu.