1 Introduction

We consider a bounded open domain \(\varOmega \) of \( {\textbf{R}} ^{n},\) with a Lipschitz-continuous boundary \(\partial \varOmega \) and \(I=\left[ 0,T\right] ,\) \(T>0.\) Our aim in this paper is to study the following parabolic p-biLaplace integrodifferential problem:

$$\begin{aligned}{} & {} \frac{\partial u\left( t,x\right) }{\partial t}+\triangle _{p}^{2}u\left( t,x\right) =f\left( t,x\right) +\int _{0}^{t}a\left( t-s\right) \triangle _{p}^{2}u\left( s,x\right) ds\quad \text {in }I\times \varOmega \end{aligned}$$
(1)
$$\begin{aligned}{} & {} u=0,\ \nabla u=0\quad \text {on }\varSigma =I\times \partial \varOmega \end{aligned}$$
(2)
$$\begin{aligned}{} & {} u\left( 0,x\right) =u_{0}\quad \text {on }\varOmega \end{aligned}$$
(3)

where

$$\begin{aligned} \triangle _{p}^{2}u:=\triangle \left( \left| \triangle u\right| ^{p-2}\triangle u\right) \end{aligned}$$
(4)

\(f\in C\left( I,L^{q}\left( \varOmega \right) \right) \) satisfies \(\left\| f\left( t\right) -f\left( t^{\prime }\right) \right\| _{L^{q}\left( \varOmega \right) }\le l\left| t-t^{\prime }\right| ,\) \(\forall t,t^{\prime }\in I\) for some strictly positive constant l. The exponent \( p> 1,\) its conjugate q satisfies \(\frac{1}{p}+\frac{1}{q}=1.\) The initial value \(u_{0}\) and a are given functions in \(W_{0}^{2,p}\left( \varOmega \right) \) and \(C\left[ 0,T\right] ,\) respectively.

During the last decades, the study of the high-order PDEs has undergone rapid development. One of our motivation for studying (1) comes from applications in area of elasticity, more precisely, it can be used in modelling of travelling waves in suspension bridges (see [13]). Other interesting applications are related to improve the visual quality of damaged and noisy images if \(1<p^{-}<2\) (see [19] and references cited therein). Note that in the stationary case and for \(p=2\) problem \(\left( 1\right) \) becomes \(\triangle ^{2}u=f\) which models the deformations of a thin homogeneous plate embedded along its beam and subjected to a distribution f of load normal to the plate. Among the most recent work concerning the parabolic p-biharmonic equation, we can review Cömert et al. [5], where the authors tried to demonstrate the existence and uniqueness of global weak solution for p-biharmonic parabolic equation with logarithmic nonlinearity. In [11], the authors got the results on blowup, extinction and non-extinction of the solutions for a nonlocal p-biharmonic parabolic equation under appropriate conditions. In [20], the author has studied p-biharmonic parabolic equation with logarithmic nonlinearity. An initial value problem for the wave p(x)-bi-Laplace with nonlinear dissipation has been considered in [21]. Using variety of techniques, the authors obtained sufficient conditions to prove the global nonexistence result. On the other hand, Roth’s time discretization is one of the most common methods for solving evolution equations, where the derivatives with respect to t are substituted by the corresponding difference quotients which eventually lead to systems of differential equations, (see [3, 4, 6, 8,9,10, 15, 16]).

The mixed finite element method is discussed for this class of partial differential equations, because of its different advantages, which are represented in the local conservation of each of (the quantity of movement, the mass, the quantity of heat, etc.) and thus the global conservation of these physical quantities. It also allows the introduction of an auxiliary variable, the so-called a Lagrange multiplier associated to a constraint that the state must satisfy, therefore obtaining a system of two equations.

In the present article, we consider a high order parabolic p-biharmonic problem with memory term. Our motivation in this choice is a good study of this type of problem by treating it analytically and numerically, using the Rothe method combined with mixed finite elements method to obtain an approximate solution of the problem (1). Some qualitative results, depending on the p values (\(1<p\le 2\) and if \(2<p<\infty \)) are proved. An optimal error estimate is discussed and supported by numerical tests.

The paper is structured as follows: We present in Sect. 2 some basic notations and material needed for our work. Section 3 is devoted to some a priori estimates and convergence results which allows us to conclude the existence of weak solution of problem (1) in the sense of definition Sect. 4. In Sect. 4 we show that for \(1< p< 2\) the problem under investigation has at most one weak solution. Section 5 contains mixed formula. The fully-discrete mixed finite element scheme and the optimal a priori error estimate for u and \(\psi \) are derived in Sect. 6. Finally, numerical tests are presented to show the effectiveness of the proposed approach.

2 Preliminaries

We define the Lebesgue space \(L^{p}\left( \varOmega \right) \) as follows

$$\begin{aligned} L^{p}\left( \varOmega \right) =\left\{ u:\varOmega \longrightarrow {\textbf{R}}, u\text { measurable and }\int _{\varOmega }\left| u\left( x\right) \right| ^{p}dx< \infty \right\} \end{aligned}$$
(5)

equipped with the norm

$$\begin{aligned} \left\| u\right\| _{L^{p}\left( \varOmega \right) }^{p}=\int _{\varOmega }\left| u\left( x\right) \right| ^{p}dx. \end{aligned}$$
(6)

Definition 1

For \(p\in [ 1,\infty [ \) and \(m\in {\textbf{N}},\) we introduce the Sobolev space

$$\begin{aligned} W^{m,p}\left( \varOmega \right) =\left\{ u\in L^{p.}\left( \varOmega \right) ;\, D^{\alpha }u\in L^{p}\left( \varOmega \right) ,\forall \alpha \in {\textbf{N}} ^{n}\text { and }\left| \alpha \right| \le m\right\} \end{aligned}$$
(7)

equipped with the following norm and semi norm

$$\begin{aligned}{} & {} \left\| u\right\| _{m,p}^{p}=\sum _{\left| \alpha \right| \le m}\left\| D^{\alpha }u\right\| _{L^{p}\left( \varOmega \right) }^{p} \end{aligned}$$
(8)
$$\begin{aligned}{} & {} \left| u\right| _{m,p}^{p}=\sum _{\left| \alpha \right| =m}\left\| D^{\alpha }u\right\| _{L^{p}\left( \varOmega \right) }^{p} \end{aligned}$$
(9)

\(W_{0}^{2,p}\left( \varOmega \right) \) is the subspace of \(W^{2,p}\left( \varOmega \right) \) which is defined by

$$\begin{aligned} W_{0}^{2,p}\left( \varOmega \right) =\left\{ u\in W^{2,p}\left( \varOmega \right) ;\, u\left| _{\partial \varOmega }\right. =0\text { and }\nabla u\left| _{\partial \varOmega }\right. =0\right\} \end{aligned}$$
(10)

Remark 1

  1. (i)

    Note that if \(p>1,\) then the spaces \(W^{2,p}\left( \varOmega \right) \) and \(W_{0}^{2,p}\left( \varOmega \right) \) are separable and reflexive Banach spaces.

  2. (ii)

    (Poincaré inequality) Let \(p\ge 1,\exists C\left( \varOmega ,p\right) \) such that

    $$\begin{aligned} \left\| u\right\| _{p}\le C\left\| \nabla u\right\| _{p},\quad \forall u\in W_{0}^{1,p}\left( \varOmega \right) . \end{aligned}$$
    (11)
  3. (iii)

    The norm \(\left\| .\right\| _{2,p}\) is equivalent to the semi norm \(\left\| \triangle \left( .\right) \right\| _{L^{p}\left( \varOmega \right) }\) over the space \(W_{0}^{2,p}\left( \varOmega \right) \) (See [1, 17]).

Lemma 1

(See [14]) Let \(x,y\in {\textbf{R}} ^{n},\) with \(x\ne y\)

  1. (1)

    For \(p\ge 2\) there exists \(C_{1}\left( p\right) \) such that

    $$\begin{aligned} \left( \left| x\right| ^{p-2}x-\left| y\right| ^{p-2}y,x-y\right) _{ {\textbf{R}} ^{n}}\ge C_{1}\left( p\right) \left| x-y\right| ^{p}. \end{aligned}$$
    (12)
  2. (2)

    For \(1< p\le 2\) there exists \(C_{2}\left( p\right) \) such that

    $$\begin{aligned} \left| \left| x\right| ^{p-2}x-\left| y\right| ^{p-2}y\right| \le C_{2}\left( p\right) \left| x-y\right| ^{p-1}. \end{aligned}$$
    (13)

Definition 2

By a weak solution of problem (1) we mean a function u satisfying:

  1. (i)
    $$\begin{aligned} u\in C\left( I,W_{0}^{-2,q}\left( \varOmega \right) \right) \cap L^{p}\left( I,W_{0}^{2,p}\left( \varOmega \right) \right) \quad \text {with }\frac{\partial u}{\partial t}\in L^{q}\left( I,W^{-2,q}\left( \varOmega \right) \right) \nonumber \\ \end{aligned}$$
    (14)
  2. (ii)
    $$\begin{aligned}{} & {} \int _{0}^{T}\int _{\varOmega }\frac{\partial u}{\partial t}vdxdt+\int _{0}^{T} \int _{\varOmega }\triangle _{p}^{2}u.vdxdt =\int _{0}^{T}\int _{\varOmega }fvdxdt\nonumber \\{} & {} \quad +\int _{0}^{T}\int _{\varOmega }\left( \int _{0}^{t}a\left( t-s\right) \triangle _{p}^{2}u\left( s,x\right) ds\right) vdxdt,\quad \forall v\in L^{p}\left( I,W_{0}^{2,p}\left( \varOmega \right) \right) .\nonumber \\ \end{aligned}$$
    (15)

3 A priori estimates and existence results

Let us divide the interval \(I=\left[ 0,T\right] \) into n subintervals where \( \tau =\frac{T}{n},\) \(t_{i}=i\tau ,\) \(u^{i}\left( x\right) =u\left( t_{i},x\right) \) and \(\delta u^{i}\left( x\right) =\frac{u^{i}\left( x\right) -u^{i-1}\left( x\right) }{\tau }\simeq \frac{\partial u}{\partial t} \) for \(i=1,\ldots ,n.\) Multiplying Eq. (1) by \(v\in W_{0}^{2,p}\left( \varOmega \right) \) and integrating over \(\varOmega \) we obtain

$$\begin{aligned}{} & {} \int _{\varOmega }\frac{\partial u}{\partial t}vdx+\int _{\varOmega }\left| \triangle u\right| ^{p-2}\triangle u\triangle vdx=\int _{\varOmega }fvdx \nonumber \\ {}{} & {} \quad +\int _{\varOmega }\left( \int _{0}^{t}a\left( t-s\right) \left| \triangle u\right| ^{p-2}\triangle u\left( s,x\right) ds\right) \triangle vdx. \end{aligned}$$
(16)

Then recurrent approximation scheme for \(i=1,\ldots ,n\) is

$$\begin{aligned}{} & {} \int _{\varOmega }\delta u^{i}vdx+\int _{\varOmega }\left| \triangle u^{i}\right| ^{p-2}\triangle u^{i}\triangle vdx=\int _{\varOmega }f^{i}vdx \nonumber \\ {}{} & {} \quad +\tau \sum _{j=0}^{i-1}a_{ij}\int _{\varOmega }\left| \triangle u^{j}\right| ^{p-2}\triangle u^{j}\triangle vdx \end{aligned}$$
(17)

where \(f^{i}\left( x\right) =f\left( t_{i},x\right) \) and \( a_{ij}=a\left( t_{i}-t_{j}\right) .\) Note that the existence of solution \( u^{i}\) at each time step \(t_{i}\) is ensured thanks to the monotonicity and coercivity of the operator \(u^{i}+\tau \triangle _{p}^{2}u^{i}-\tau ^{2}\triangle _{p}^{2}u^{i-1}.\)

Theorem 1

Let \(p> 2.\) Then the problem (1) admits a weak solution u in the sense of Definition 1.

Before we proceed to prove Theorem 1, we need some auxiliary lemmas.

Lemma 2

There exists a positive constant C independent of n such that

$$\begin{aligned} \left\| u^{k}\right\| _{L^{2}\left( \varOmega \right) }\le & {} C,\quad k=1,\ldots ,n \\ \sum _{i=1}^{k}\tau \left\| \triangle u^{i}\right\| _{L^{p}\left( \varOmega \right) }^{p}\le & {} C,\quad k=1,\ldots ,n. \end{aligned}$$

Proof

Testing with \(u^{i}\) into (17) and summing over \(i=1,\ldots ,k,\) we get

$$\begin{aligned}{} & {} \sum _{i=1}^{k}\int _{\varOmega }\left( u^{i}-u^{i-1}\right) u^{i}dx+\tau \sum _{i=1}^{k}\int _{\varOmega }\left| \triangle u^{i}\right| ^{p}dx =\tau \sum _{i=1}^{k}\int _{\varOmega } f^{i}u^{i}dx\nonumber \\{} & {} \quad +\,\tau ^{2}\sum _{i=1}^{k}\sum _{j=0}^{i-1}a_{ij}\int _{\varOmega }\left| \triangle u^{j}\right| ^{p-2}\triangle u^{j}\triangle u^{i}dx. \end{aligned}$$
(18)

We estimate each term of (18) separately, we obtain

$$\begin{aligned} \sum _{i=1}^{k}\int _{\varOmega }\left( u^{i}-u^{i-1}\right) u^{i}dx=\frac{1}{2} \left\| u^{k}\right\| _{L^{2}\left( \varOmega \right) }^{2}-\frac{1}{2} \left\| u_{0}\right\| _{L^{2}\left( \varOmega \right) }^{2}+\frac{1}{2} \sum _{i=1}^{k}\left\| u^{i}-u^{i-1}\right\| _{L^{2}\left( \varOmega \right) }^{2}. \end{aligned}$$
(19)

Using Holder inequality and \(\epsilon \)-Young inequality we have

$$\begin{aligned} \tau \left| \sum _{i=1}^{k}\int _{\varOmega }f^{i}u^{i}dx\right|\le & {} \tau \sum _{i=1}^{k}\left( \int _{\varOmega }\left| f^{i}\right| ^{q}dx\right) ^{\frac{1}{q}}\left( \int _{\varOmega }\left| u^{i}\right| ^{p}dx\right) ^{\frac{1}{p}} \nonumber \\\le & {} \sum _{i=1}^{k}\tau \frac{C}{\epsilon }\int _{\varOmega }\left| f^{i}\right| ^{q}dx+\sum _{i=1}^{k}\tau C\varepsilon \int _{\varOmega }\left| u^{i}\right| ^{p}dx. \end{aligned}$$
(20)

Taking into account the continuous embedding of \(W_{0}^{2,p}\left( \varOmega \right) \) into \(L^{p}\left( \varOmega \right) \) we arrive at

$$\begin{aligned} \tau \left| \sum _{i=1}^{k}\int _{\varOmega }f^{i}u^{i}dx\right|\le & {} C\sum _{i=1}^{k}\tau +C\epsilon \sum _{i=1}^{k}\tau \int _{\varOmega }\left| \triangle u^{i}\right| ^{p}dx \nonumber \\\le & {} C+C\epsilon \tau \sum _{i=1}^{k}\left\| \triangle u^{i}\right\| _{L^{p}\left( \varOmega \right) }^{p}. \end{aligned}$$
(21)

For the memory term we proceed as follows

$$\begin{aligned}{} & {} \left| \tau ^{2}\sum _{i=1}^{k}\sum _{j=0}^{i-1}a_{ij}\int _{\varOmega }\left| \triangle u^{j}\right| ^{p-2}\triangle u^{j}\triangle u^{i}dx\right| \le \tau ^{2}\sum _{i=1}^{k}\sum _{j=0}^{i-1}C\int _{\varOmega }\left| \triangle u^{j}\right| ^{p-1}\left| \triangle u^{i}\right| dx\nonumber \\{} & {} \quad \le \tau ^{2}\sum _{i=1}^{k}\sum _{j=0}^{i-1}C\left( \int _{\varOmega }\left| \triangle u^{j}\right| ^{\left( p-1\right) q}dx\right) ^{ \frac{1}{q}}\left( \int _{\varOmega }\left| \triangle u^{i}\right| ^{p}dx\right) ^{\frac{1}{p}} \nonumber \\{} & {} \quad \le \tau ^{2}\sum _{i=1}^{k}\sum _{j=0}^{i-1}\frac{C}{\epsilon } \int _{\varOmega }\left| \triangle u^{j}\right| ^{p}dx+\sum _{i=1}^{k}i\tau ^{2}C\epsilon \int _{\varOmega }\left| \triangle u^{i}\right| ^{p}dx \nonumber \\{} & {} \quad \le C\sum _{i=1}^{k}\sum _{j=0}^{i-1}\tau ^{2}\left\| \triangle u^{j}\right\| _{L^{p}\left( \varOmega \right) }^{p}+C\epsilon \tau \sum _{i=1}^{k}\left\| \triangle u^{i}\right\| _{L^{p}\left( \varOmega \right) }^{p}. \end{aligned}$$
(22)

Substituting (19), (21) and (22) into (18) we see

$$\begin{aligned}{} & {} \left\| u^{k}\right\| _{L^{2}\left( \varOmega \right) }^{2}+\left( 1-C\epsilon \right) \tau \sum _{i=1}^{k}\left\| \triangle u^{i}\right\| _{L^{p}\left( \varOmega \right) }^{p}\le \left\| u^{0}\right\| _{L^{2}\left( \varOmega \right) }^{2}\nonumber \\ {}{} & {} \quad +\,C\sum _{i=1}^{k}\sum _{j=0}^{i-1}\tau ^{2}\left\| \triangle u^{j}\right\| _{L^{p}\left( \varOmega \right) }^{p}+C. \end{aligned}$$
(23)

Now, choosing \(\epsilon \) such that \(\left( 1-C\epsilon \right) > 0\) and making use of Gronwall lemma we arrive at the desired result. \(\square \)

Lemma 3

The following estimate holds

$$\begin{aligned} \left\| \partial _{t}u^{n}\right\| _{W^{-2,q}\left( \varOmega \right) }^{2}\le C. \end{aligned}$$
(24)

Proof

Note that the above notations allows us to rewrite (17) as

$$\begin{aligned}{} & {} \int _{\varOmega }\partial _{t}u^{n}\left( t\right) vdx+\int _{\varOmega }\triangle _{p}^{2}\overline{u^{n}}\left( t\right) vdx=\int _{\varOmega }f^{n}\left( t\right) vdx \nonumber \\ {}{} & {} \qquad +\tau \sum _{j=0}^{i-1}a_{ij}\int _{\varOmega }\triangle _{p}^{2}u^{j}vdx,\forall v\in W_{0}^{2,p}\left( \varOmega \right) , \end{aligned}$$
(25)

where \(f^{n}\left( t\right) =f^{i},\) \(t\in \left[ t_{i-1},t_{i}\right] ,\) \( 1\le i\le n.\)

Now, taking into account that for \(p> 2,\) \(\triangle _{p}^{2}:W_{0}^{2,p}\left( \varOmega \right) \longrightarrow W^{-2,q}\left( \varOmega \right) \) is bounded (see [7]), Holder inequality gives us

$$\begin{aligned} \left| \int _{\varOmega }\partial _{t}u^{n}\left( t\right) vdx\right|\le & {} \left| \int _{\varOmega }f^{n}vdx\right| +\tau \sum _{j=0}^{i-1}a_{ij}\int _{\varOmega }\left| \triangle u^{j}\right| ^{p-1}\left| \triangle v\right| dx+\left| \int _{\varOmega } \triangle _{p}^{2}\overline{u^{n}}\left( t\right) vdx\right| \nonumber \\\le & {} \left\| f^{n}\right\| _{L^{q}\left( \varOmega \right) }\left\| v\right\| _{L^{p}\left( \varOmega \right) }+\tau C\sum _{j=0}^{i-1}\left( \int _{\varOmega }\left| \triangle u^{j}\right| ^{\left( p-1\right) q}dx\right) ^{\frac{1}{q}}\left( \int _{\varOmega }\left| \triangle v\right| ^{p}dx\right) ^{\frac{1}{p}} \nonumber \\{} & {} +C\left\| v\right\| _{L^{p}\left( \varOmega \right) } \nonumber \\\le & {} C\left\| f\right\| _{C\left( I,L^{q}\left( \varOmega \right) \right) }\left\| \triangle v\right\| _{L^{p}\left( \varOmega \right) }+\tau C\sum _{j=0}^{i-1}\left\| \triangle u^{j}\right\| _{L^{p}\left( \varOmega \right) }^{\frac{p}{q}}\left\| \triangle v\right\| _{L^{p}\left( \varOmega \right) } \nonumber \\{} & {} +C\left\| \triangle v\right\| _{L^{p}\left( \varOmega \right) } \end{aligned}$$
(26)

\(\epsilon \)-Young inequality implies

$$\begin{aligned} \left| \int _{\varOmega }\partial _{t}u^{n}\left( t\right) vdx\right| \le C\left\| \triangle v\right\| _{L^{p}\left( \varOmega \right) }+\left( \sum _{j=0}^{i-1}\frac{C}{\epsilon }\tau \left\| \triangle u^{j}\right\| _{L^{p}\left( \varOmega \right) }^{p}+\sum _{j=0}^{i-1}C\epsilon \tau \left| \varOmega \right| \right) \left\| \triangle v\right\| _{L^{p}\left( \varOmega \right) }. \end{aligned}$$
(27)

Now choosing \(\epsilon =1\) and making use of the Lemma 1, we have

$$\begin{aligned} \left| \int _{\varOmega }\partial _{t}u^{n}\left( t\right) vdx\right| \le C\left\| \triangle v\right\| _{L^{p}\left( \varOmega \right) },\quad \forall v\in W_{0}^{2,p}\left( \varOmega \right) . \end{aligned}$$
(28)

Hence,

$$\begin{aligned} \left\| \partial _{t}u^{n}\left( t\right) \right\| _{W^{-2,q}\left( \varOmega \right) }=\sup _{v\in W_{0}^{2,p}\left( \varOmega \right) } \frac{\left| \int _{\varOmega }\partial _{t}u^{n}\left( t\right) vdx\right| }{\left\| \triangle v\right\| _{L^{p}\left( \varOmega \right) }}\le C. \end{aligned}$$
(29)

This completes the proof. \(\square \)

Proof of Theorem 1

From Lemmas 2 and 3 we can deduce

$$\begin{aligned}{} & {} \left\| \partial _{t}u^{n}\right\| _{L^{q}\left( I,W^{-2,q}\left( \varOmega \right) \right) }\le C \end{aligned}$$
(30)
$$\begin{aligned}{} & {} \left\| \overline{u^{n}}\right\| _{L^{p}\left( I,W_{0}^{2,p}\left( \varOmega \right) \right) }^{p}=\int _{0}^{T}\left\| \triangle \overline{u^{n} }\left( t\right) \right\| _{L^{p}\left( \varOmega \right) }^{p}dt\le C \end{aligned}$$
(31)

and

$$\begin{aligned} \left\| \overline{u^{n}}\right\| _{C\left( I,L^{2}\left( \varOmega \right) \right) }\le C \end{aligned}$$
(32)

then, Lemma 1.3.13 in [12] implies that there exists \(u\in C \left( I,W^{-2,q}\left( \varOmega \right) \right) \cap L^{p}\left( I,W_{0}^{2,p}\left( \varOmega \right) \right) \) having \(\partial _{t}u\in L^{q}\left( I,W^{-2,q}\left( \varOmega \right) \right) \) and a subsequence of \(u^{n}\) again denoted \(u^{n}\) satisfying

$$\begin{aligned} u^{n}\longrightarrow & {} u\quad \text {in }C\left( I,W^{-2,q}\left( \varOmega \right) \right) \nonumber \\ u^{n}\left( t\right)&\rightharpoonup&u\left( t\right) \quad \text {in } W_{0}^{2,p}\left( \varOmega \right) \nonumber \\ \overline{u^{n}}&\rightharpoonup&u\quad \text {in }L^{p}\left( I,W_{0}^{2,p}\left( \varOmega \right) \right) \nonumber \\ \partial _{t}u^{n}&\rightharpoonup&\partial _{t}u\quad \text {in }L^{q}\left( I,W^{-2,q}\left( \varOmega \right) \right) \nonumber \\ f^{n}&\rightharpoonup&f\quad \text {in }L^{q}\left( I,L^{q}\left( \varOmega \right) \right) \end{aligned}$$
(33)

as \(n\longrightarrow \infty .\)

On the other hand, we know that \(\triangle _{p}^{2}:W_{0}^{2,p}\left( \varOmega \right) \longrightarrow W^{-2,q}\left( \varOmega \right) \) is hemicontinuous operator see [7]. Using this fact we get

$$\begin{aligned}{} & {} \int _{\varOmega }\triangle _{p}^{2}\overline{u^{n}}\left( t\right) vdx\longrightarrow \int _{\varOmega }\triangle _{p}^{2}u\left( t\right) vdx=\int _{\varOmega }\left| \triangle u\left( t\right) \right| ^{p-2}\triangle u\left( t\right) \triangle vdx,\nonumber \\ {}{} & {} \quad \forall v\in W_{0}^{2,p}\left( \varOmega \right) \end{aligned}$$
(34)

for \(n\longrightarrow \infty .\)

We proceed similarly as in [2] we can easily check that

$$\begin{aligned}{} & {} \tau \sum _{j=0}^{i-1}a_{ij}\int _{\varOmega }\triangle _{p}^{2}u^{j}vdx\rightharpoonup \int _{0}^{t}a\left( t-s\right) \triangle _{p}^{2}u\left( s,x\right) ds\text { in }L^{q}\left( I,W^{-2,q}\left( \varOmega \right) \right) \nonumber \\ {}{} & {} \quad \text {for }\tau \longrightarrow 0. \end{aligned}$$
(35)

Thanks to assumptions on f;  property (33)\(_5\) is a consequence of the estimate

$$\begin{aligned} \left\| f^{n}\left( t\right) -f\left( t\right) \right\| _{L^{q}\left( \varOmega \right) }\le \frac{C}{n}. \end{aligned}$$
(36)

Now, integrating Eq. (25) from 0 to T yields

$$\begin{aligned}{} & {} \int _{0}^{T}\int _{\varOmega }\partial _{t}u^{n}\left( t\right) vdx+\int _{0}^{T}\int _{\varOmega }\triangle _{p}^{2}\overline{u^{n}}\left( t\right) vdx=\int _{0}^{T}\int _{\varOmega } f^{n}vdx \nonumber \\{} & {} \quad +\,\tau \sum _{j=0}^{i-1}a_{ij}\int _{0}^{T}\int _{\varOmega }\triangle _{p}^{2}u^{j}vdx,\quad \forall v\in W_{0}^{2,p}\left( \varOmega \right) . \end{aligned}$$
(37)

Taking limit as \(n\rightarrow \infty \) in (37), we get

$$\begin{aligned}{} & {} \int _{0}^{T}\int _{\varOmega }\frac{\partial u}{\partial t}vdxdt+\int _{0}^{T} \int _{\varOmega }\triangle _{p}^{2}uvdxdt =\int _{0}^{T}\int _{\varOmega }fvdxdt\nonumber \\{} & {} \quad +\int _{0}^{T}\int _{\varOmega }\left( \int _{0}^{t}a\left( t-s\right) \triangle _{p}^{2}u\left( s,x\right) ds\right) vdxdt,\forall v \in W_{0}^{2,p}\left( \varOmega \right) . \end{aligned}$$
(38)

Therefore concluding the proof.

4 Uniqueness of the weak solution

Theorem 2

Let \(1< p\le 2.\) Then problem (1) has at most one solution.

Proof

Suppose that problem (1) has two solutions \(u_{1}\) and \(u_{2}.\) Subtracting the Eq. (16) for \(u_{2}\) from the Eq. (16) for \(u_{1},\) we get

$$\begin{aligned}{} & {} \int _{\varOmega }\partial _{t}\left( u_{1}-u_{2}\right) vdx+\int _{\varOmega }\left( \left| \triangle u_{1}\right| ^{p-2}\triangle u_{1}-\left| \triangle u_{2}\right| ^{p-2}\triangle u_{2}\right) \triangle vdx \nonumber \\{} & {} \quad =\int _{\varOmega }\left( \int _{0}^{t}a\left( t-s\right) \left( \left| \triangle u_{1}\right| ^{p-2}\triangle u_{1}-\left| \triangle u_{2}\right| ^{p-2}\triangle u_{2}\right) ds\right) \triangle vdx. \end{aligned}$$
(39)

Choosing \(v=u_{1}-u_{2}\) in (39) and making use of Lemma 1 we obtain

$$\begin{aligned}{} & {} \frac{d}{dt}\left\| u_{1}-u_{2}\right\| _{L^{2}\left( \varOmega \right) }^{2}+C_{1}\left\| \triangle \left( u_{1}-u_{2}\right) \right\| _{L^{p}\left( \varOmega \right) }^{p} \nonumber \\{} & {} \quad \le \int _{\varOmega }\left( \int _{0}^{t}a\left( t-s\right) \left( \left| \triangle u_{1}\right| ^{p-2}\triangle u_{1}-\left| \triangle u_{2}\right| ^{p-2}\triangle u_{2}\right) ds\right) \triangle \left( u_{1}-u_{2}\right) dx. \end{aligned}$$
(40)

Integrating (40) over \(\left[ 0,\xi \right] \) for \(\xi \in \left[ 0,T\right] \) and taking into account that \(u_{1}\left( 0\right) =u_{2}\left( 0\right) \) we have

$$\begin{aligned}{} & {} \left\| u_{1}\left( \xi \right) -u_{2}\left( \xi \right) \right\| _{L^{2}\left( \varOmega \right) }^{2}+C_{1}\int _{0}^{\xi }\left\| \triangle \left( u_{1}-u_{2}\right) \right\| _{L^{p}\left( \varOmega \right) }^{p}dt\nonumber \\{} & {} \quad \le \int _{0}^{\xi }\int _{\varOmega }\left( \int _{0}^{t}a\left( t-s\right) \left( \left| \triangle u_{1}\right| ^{p-2}\triangle u_{1}-\left| \triangle u_{2}\right| ^{p-2}\triangle u_{2}\right) ds\right) \triangle \left( u_{1}-u_{2}\right) dxdt. \end{aligned}$$
(41)

Applying \(\epsilon \)-Young inequality to the right hand side of (41) we get

$$\begin{aligned} C_{1}\int _{0}^{\xi }\left\| \triangle \left( u_{1}-u_{2}\right) \right\| _{L^{p}\left( \varOmega \right) }^{p}dt\le & {} \frac{C}{\epsilon } \int _{0}^{\xi }\int _{\varOmega }\left( \int _{0}^{t}a\left( t-s\right) \left( \left| \triangle u_{1}\right| ^{p-2}\triangle u_{1}-\left| \triangle u_{2}\right| ^{p-2}\triangle u_{2}\right) ds\right) ^{q}dxdt\nonumber \\{} & {} +C\epsilon \int _{0}^{\xi }\left\| \triangle \left( u_{1}-u_{2}\right) \right\| _{L^{p}\left( \varOmega \right) }^{p}dt. \end{aligned}$$
(42)

Now, by virtue of Lemma 1 and Holder inequality, the first term in the right hand side of (42) satisfies

$$\begin{aligned}{} & {} \int _{0}^{t}a\left( t-s\right) \left( \left| \triangle u_{1}\right| ^{p-2}\triangle u_{1}-\left| \triangle u_{2}\right| ^{p-2}\triangle u_{2}\right) ds \nonumber \\{} & {} \quad \le \left\| a\right\| _{C\left( I\right) }\left( \int _{0}^{t}ds\right) ^{\frac{1}{p}}\left( \int _{0}^{t}\left| \left| \triangle u_{1}\right| ^{p-2}\triangle u_{1}-\left| \triangle u_{2}\right| ^{p-2}\triangle u_{2}\right| ^{q}ds\right) ^{\frac{1}{q}} \nonumber \\{} & {} \quad \le \left\| a\right\| _{C\left( \left[ 0,T\right] \right) }T^{\frac{ 1}{p}}C_{2}\left( \int _{0}^{t}\left| \triangle \left( u_{1}-u_{2}\right) \right| ^{\left( p-1\right) q}ds\right) ^{\frac{1}{q}} \nonumber \\{} & {} \quad \le C\left( \int _{0}^{t}\left| \triangle \left( u_{1}-u_{2}\right) \right| ^{p}ds\right) ^{\frac{1}{q}}. \end{aligned}$$
(43)

Substituting (43) into (42) and choosing \( \epsilon \) such that \(C_{1}-C\epsilon > 0\) we arrive at

$$\begin{aligned} \left( C_{1}-C\epsilon \right) \int _{0}^{\xi }\int _{\varOmega }\left| \triangle \left( u_{1}-u_{2}\right) \right| ^{p}dxdt\le C\int _{0}^{\xi }\int _{\varOmega }\int _{0}^{t}\left| \triangle \left( u_{1}-u_{2}\right) \right| ^{p}dsdxdt. \end{aligned}$$
(44)

Gronwall lemma shows that

$$\begin{aligned} \left\| u_{1}-u_{2}\right\| _{L^{p}\left( \left[ 0,\xi \right] ,W_{0}^{2,p}\left( \varOmega \right) \right) }^{p}=\int _{0}^{\xi }\int _{\varOmega }\left| \triangle \left( u_{1}-u_{2}\right) \right| ^{p}dxdt=0,\quad \forall \xi \in \left[ 0,T\right] . \end{aligned}$$
(45)

Consequently,

$$\begin{aligned} u_{1}=u_{2}\quad \text {in }L^{p}\left( \left[ 0,T\right] ,W_{0}^{2,p}\left( \varOmega \right) \right) . \end{aligned}$$
(46)

\(\square \)

Remark 2

Multiplying Eq. (1) by \(a\left( t-s\right) \) and integrating over \(\left[ 0,t\right] ,\) we obtain

$$\begin{aligned} \int _{0}^{t}a\left( t-s\right) \triangle _{p}^{2}u\left( t,x\right) ds= & {} \int _{0}^{t}a\left( t-s\right) f\left( t,x\right) ds+\int _{0}^{t}a\left( t-s\right) \int _{0}^{s}a\left( s-\mu \right) \triangle _{p}^{2}u\left( \mu ,x\right) d\mu ds \nonumber \\{} & {} -a\left( 0\right) u\left( t,x\right) +a\left( t\right) u_{0}\left( x\right) +\int _{0}^{t}a^{\prime }\left( t-s\right) u\left( t,x\right) ds. \end{aligned}$$
(47)

Now, let \(M\left( t,x\right) =\int _{0}^{t}a\left( t-s\right) \triangle _{p}^{2}u\left( t,x\right) ds.\) It is clear that

$$\begin{aligned} M\left( t,x\right) =\int _{0}^{t}a\left( t-s\right) M\left( s,x\right) ds+F\left( t,x,u\right) \end{aligned}$$
(48)

where

$$\begin{aligned} F\left( t,x,u\right) =\int _{0}^{t}a\left( t-s\right) f\left( t,x\right) ds-a\left( 0\right) u\left( t,x\right) +a\left( t\right) u_{0}\left( x\right) +\int _{0}^{t}a^{\prime }\left( t-s\right) u\left( t,x\right) ds. \end{aligned}$$
(49)

Therefore, problem (1) takes the form of the following system

$$\begin{aligned} \left\{ \begin{array}{l} \frac{\partial u\left( t,x\right) }{\partial t}+\triangle _{p}^{2}u\left( t,x\right) =f\left( t,x\right) +M\left( t,x\right) \quad \text {in }I\times \varOmega \\ M\left( t,x\right) =\int _{0}^{t}a\left( t-s\right) M\left( s,x\right) ds+F\left( t,x,u\right) \quad \text {in }I\times \varOmega \\ u=0,\nabla u=0\quad \text {on }\varSigma =I\times \partial \varOmega \\ u\left( 0,x\right) =u_{0}\quad \text {on }\varOmega \\ M\left( 0,x\right) =0\quad \text {on }\varOmega . \end{array} \right. \end{aligned}$$
(50)

Which may be investigated in another occasion.

5 Mixed formulation

In this paper, we propose an analysis of the mixed formula, taking into account the following observation:

that if \(\varPhi (w)=|w|^{p-2}w,\) the inverse is specified as

$$\begin{aligned} \varPhi ^{-1}(w)=sgn(w)|w|^{{1}\over {p-1}}w=|w|^{q-2}w, \end{aligned}$$

depending on this, we choose the following auxiliary variable

$$\begin{aligned} \psi ^i=|\triangle u^i|^{p-2}\triangle u^i. \end{aligned}$$
(51)

Taking \(V=W^{2,p}_0(\varOmega )\) and \(W=L^q(\varOmega ),\) we rewrite problem (1) as follows

$$\begin{aligned} \left\{ \begin{array}{l} -\triangle u^i=|\psi ^i|^{q-2}\psi ^i,\\ -\triangle \psi ^i=-f^i+\delta u^i-\tau \sum _{j=1}^{i-1}a_{ij}\varDelta \psi ^i, \end{array} \right. \end{aligned}$$
(52)

we also write the mixed formulation of (52) as: we wish to determine a pair \((u^i,\psi ^i)\in V\times W\) such that

$$\begin{aligned} \left\{ \begin{array}{l} a(\psi ^i,v)-b(u^i,v)=0\quad \forall v\in V,\\ b(\psi ^i,\varphi )=L_Y(\varphi )\quad \forall \varphi \in W, \end{array} \right. \end{aligned}$$
(53)

where

$$\begin{aligned}{} & {} a(\psi ^i,v):=\big (|\psi ^i|^{q-2}\psi ^i,v\big ), \end{aligned}$$
(54)
$$\begin{aligned}{} & {} b(\psi ^i,\varphi ):=-\big (\triangle \psi ^i,\varphi \big ), \end{aligned}$$
(55)
$$\begin{aligned}{} & {} L_Y(\varphi ):=\big ((-f^i+\delta u^i),\varphi \big )+\tau \sum _{j=1}^{i-1}a_{ij}b(\psi ^j,\varphi ), \end{aligned}$$
(56)

where \(f^i=f(t_i,x).\)

Proposition 1

(Inf-sup condition) There exists \(\gamma ,\) \(C\ge 0\) such that for \(u\in V,\) we have

$$\begin{aligned} \gamma \le C \inf _{0\ne \varphi \in W}\sup _{0\ne u^i\in V} {{b(u^i,\varphi )}\over {\Vert u^i\Vert _V\Vert \varphi \Vert _{W}}}. \end{aligned}$$
(57)

Proof

See [6].

6 Full discretization

Let \(\varUpsilon _T\) be a partition of into disjoint triangles T which means that \({\bar{\varOmega }}=\cup _{T\in \varUpsilon _T} {\bar{T}}\) and also for each T,  K \(\in \varUpsilon _T,\) we have that \(T\cap K\) is either: a vertex, an edge, a face, or the whole of K and T.

Noting that this triangulation is regular to the concept of Ciarlet, where the shape regularity constant of T is given as follows

$$\begin{aligned} \exists \mu >0;\,\mu =\inf _{T\in \varUpsilon _T}{{h_T}\over {\rho _T}}\ \quad \forall T\in \varUpsilon _h, \end{aligned}$$
(58)

with \(\rho _T\) is the diameter of the largest ball contained inside T and \(h_T\) is the diameter of T.

Let \({\textbf{P}}^k(\varUpsilon _h)\) express the space of piecewise polynomials of degree k over the triangulation \(\varUpsilon _h\)

$$\begin{aligned} {\textbf{P}}^k(\varUpsilon _h)=\big \{\phi :\,\phi _{\setminus T}\in {\textbf{P}}^k(T),\,\,\forall T\in \varUpsilon _h \big \}. \end{aligned}$$
(59)

The discrete finite spaces is defined as follows

$$\begin{aligned} V^h={\textbf{P}}^k(\varUpsilon _h)\cap C^0({\bar{\varOmega }}), \end{aligned}$$
(60)

and

$$\begin{aligned} V^h_0=\big \{\phi \in V^h;\,\phi _{\backslash \partial \varOmega }=0\big \}, \end{aligned}$$
(61)

where R is the Ritz projection operator such that

$$\begin{aligned} \int _\varOmega \nabla (R v)\nabla \phi =\int _\varOmega \nabla v\nabla \phi dx,\quad \forall \phi \in V^h\cap H^1_0(\varOmega ). \end{aligned}$$
(62)

We give the mesh size h as

$$\begin{aligned} h=\max _{T\in \varUpsilon _T} h_T. \end{aligned}$$
(63)

Then, the fully-discrete mixed finite element scheme for (53) is written as: find a pair \((u^i_h,\psi ^i_h)\in V^h_0\times V^h\)

$$\begin{aligned} \left\{ \begin{array}{l} a(\psi ^i_h,v)-b_h(u^i_h,v)=0,\\ b_h(\psi ^i_h,\varphi )=L(\varphi )\quad \forall (v,\varphi )\in V^h\times V^h_0, \end{array} \right. \end{aligned}$$
(64)

from Green formulation, we have

$$\begin{aligned} b_h(u_h,v)= \sum _{T\in \varUpsilon _h}\int _T \nabla u_h\nabla vdx=\int _\varOmega \nabla u_h\nabla vdx, \end{aligned}$$
(65)

by substituting (65) in (64), Then the problem (64) can be written as: find a pair \((u^i_h,\psi ^i_h)\in V^h_0\times V^h\)

$$\begin{aligned} \left\{ \begin{array}{l} \int _\varOmega |\psi ^i_h|^{q-2}\psi ^i_hvdx-\int _\varOmega \nabla u_h^i\nabla vdx=0,\\ \int _\varOmega \nabla \psi ^i_h\nabla \varphi dx=\int _\varOmega (f^i-\delta u^i)\varphi dx+\sum _{j=1}^{i-1}a_{ij}\int _\varOmega \nabla \psi ^j\nabla \varphi dx\quad \forall (v,\varphi )\in V^h\times V^h_0. \end{array} \right. \end{aligned}$$
(66)

Proceeding as in [6] and making use of properties of a(., .) (see [18, prop. 3.1]) and Lemma 01 in [6], we can arrive at the following corollary.

Theorem 3

For \(m\ge 2,\) there exists \(C\ge 0\) such that

$$\begin{aligned}{} & {} \Vert u^i-u^i_h\Vert ^{p-1}_{W^{2,p}_h(\varOmega )}+\Vert w^i-w^i_h\Vert _{L^q(\varOmega )}\le C\big (h^{{q\over 2}(m+1)}|w^i|^{q\over 2}_{W^{m+1,q}(\varOmega )}+h^{m+1}|w^i|_{W^{m+1,q}(\varOmega )}\\{} & {} \quad +\,h^{m-1}|u^i|_{W^{m+1,p}(\varOmega )}+h^{m+1}|\delta u^i|_{W^{m+1,q}(\varOmega )}\big ). \end{aligned}$$

7 Numerical experiment

In this section, we focus on a numerical experiment that shows the efficiency and precision of Galerkin mixed finite element method.

In this experiment, the unknown function u(t, x, y) and the auxiliary variable \(\psi (t,x,y)\) are approximated by a linear polynomial, that is, \(r=k=1.\)

The function a in the memory term is chosen as \(a(t,s)=1.\)

Example 1

In this test, we describe the computational domain \(\varOmega = (0, 1)\) and the interval is(0, 1). For this test example we take the step length \(h\in \{{1\over 3},{1\over 6},{1\over 12},{1\over 24},{1\over 48},{1\over 96}\}.\) Numerically errors are calculated at final time level \(t_i=1\) with \(\tau =10.\)

The right hand side f(t, x) and the auxiliary variable \(\psi (t,x)\) are chosen in such a way that the exact solution is \(u(t,x)={{1\over {2\pi }}\cos ({\pi \over 2} x)te^{-t}}.\) Graph of convergence of error of u and \(\psi \) in \(W^{2,p}_0(\varOmega )\) and \(L^q(\varOmega ),\) is given in Fig. 1 and numerical and exact solutions are plotted in Fig. 2.

Fig. 1
figure 1

The results of error for u and \(\psi \) in log log-plot at \(t=1\) for \(p=3,4,5\) respectively

Full size image
Fig. 2
figure 2

The Surfac for \(u^i_h\) and \(\psi ^i_h\) on \([0,1]\times [0,1]\) for \(p=4\)

Full size image
Fig. 3
figure 3

The results of error for u and \(\psi \) in log log-plot at \(t=2^5\tau \) for \(p=2.5,3,4\) respectively

Full size image
Fig. 4
figure 4

The Surfac for \(u^i_h\) and \(\psi ^i_h\) on \([0,1]\times [0,1]\) at \(t_i=2^5\tau \) for \(p=4\)

Full size image

Example 2

For this test example, the computational domain is the rectangle \(\varOmega =\{(x, y):\,(x, y)\in (0, 1)\times (0, 1)\}\) and time interval is taken to be (0, 1). We take the step length \(h\in \{{1\over 3},{1\over 6},{1\over 12},{1\over 24},{1\over 48}\}\) and \(p=3,4,5.\) Numerically errors are calculated at final time level \(t_i=2^5\tau \) with \(\tau =2^7.\)

The right hand side f(t, x, y) and the auxiliary variable \(\psi (t,x,y)\) are chosen in such a way that the exact solution is \(u(t,x,y)={{1\over {\pi ^2}}\cos ({\pi \over 2} x)\cos ({\pi \over 2} y)te^{-t}}.\) Graph of convergence of error of u and \(\psi \) in \(W^{2,p}_0(\varOmega )\) and \(L^q(\varOmega ),\) is given in Fig. 3 and numerical and exact solutions are plotted in Fig. 4.