Abstract
In this paper, we consider a wave equation on a bounded domain with a Sturm–Liouville operator with a singular intermediate coefficient and a singular potential. To obtain and evaluate the solution, the method of separation of variables is used, then the expansion in the Fourier series in terms of the eigenfunctions of the Sturm–Liouville operator is used. The Sturm–Liouville eigenfunctions are determined by such coefficients using the modified Prufer transform. Existence, uniqueness and consistency theorems are also proved for a very weak solution of the wave equation with singular coefficients.
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1 Introduction
The purpose of this work is to establish the results on the well-posedness of the wave equation for the Sturm–Liouville operator with a singular intermediate coefficient and a singular potential.
In [22], very weak solutions of the wave equation for the Landau Hamiltonian with an irregular electromagnetic field were obtained for an unbounded domain in the space. A number of works [1,2,3,4,5,6,7,8,9, 12, 14, 15, 17, 19, 20, 23] are also devoted to this topic. The difference between our results is that we consider the problem in a bounded domain. We have obtained similar results in the work [21], so our current work is a further development of these results, allowing one to include the intermediate term.
It is well known that the wave equation is easily reduced to ordinary linear equations by the “separation of variables" method (see, for example, [13]).
To obtain the main results, we present some information about the Sturm–Liouville operator with singular potentials. Savchuk and Shkalikov in [25] obtained eigenvalues and eigenfunctions of the Sturm–Liouville operator with singular potentials. This method was further developed in the works [18, 24, 26, 27]. We are guided by this method and will develop it with the addition of an intermediate coefficient, and, accordingly, additional conditions will be imposed on the coefficients, and the regularity requirements will be relaxed.
In particular, we consider the problem of constructing eigenvalues and eigenfunctions of the Sturm–Liouville operator \({\mathcal {L}}\) generated on the interval (0,1) by the differential expression
with the boundary conditions
We first assume that \(p \in W^2_1(0,1)\) (p and its first derivative are in \(L^2\)), and that the potential q is of the form
We consider the eigenvalue equation \({\mathcal {L}}y=\lambda y\). Introducing the substitution
we get the equation
while the boundary conditions do not change:
We introduce the quasi-derivative in the following form
then Eq. (1.5) transforms to the equation
We introduce
then we pass from the (1.7) to the system
We make the substitution
which is a modification of the Prufer substitution [16]. Here we have
The solution of Eq. (1.8) will be sought in the form \(\theta (x,\lambda )=\lambda ^\frac{1}{2} x+\eta (x,\lambda ), \) where
Using the method of successive approximations, it is easy to show that this equation has a solution that is uniformly bounded for \(0\le x\le 1\) and \(\lambda \ge 1\), namely,
Since \(p\in W^1_2(0,1), \, \nu \in L^2(0,1)\) and \(\nu ^2\in L^1(0,1)\), by virtue of the Riemann-Lebesgue lemma \(\eta (x,\lambda )=o(1)\) at \(\lambda \rightarrow \infty \). Therefore, we have
moreover \(\theta (0,\lambda )=0.\)
Using the Riemann-Lebesgue lemma again, from Eq. (1.9) we find
Using the boundary conditions (1.6) we obtain
Then the eigenvalues of Eq. (1.5) with the boundary conditions (1.6) are given by
and the corresponding eigenfunctions are
The first derivatives of \(\tilde{\psi _n}\) are then given by the formulas
Let us estimate \(\Vert {\tilde{\psi }}_n\Vert _{L^2}\) using the formula (1.11) as follows
since \(\nu \in L^2(0,1)\), \(p\in W^2_1(0,1)\) and \(\lambda _n\rightarrow \infty \) at \(n\rightarrow \infty \).
Also, according to Theorem 4 in [25], we have
for sufficiently large n, it means that there exist some \(C_0>0\), such that \(C_0<\Vert {\tilde{\phi }}_n\Vert _{L^2}<\infty \). Since the eigenfunctions (1.11) form an orthogonal basis in \(L^2(0,1)\), we normalise them for further use
Returning again to the substitution (1.4), we obtain the eigenfunctions
of the operator \({\mathcal {L}}\) generated by the differential expression (1.1) with the boundary conditions (1.2). In this case, the eigenvalues remain as in (1.10). It should be noted that the eigenfunctions \(\phi _n\) are orthogonal in the weighted space \(L^2_g(0,1)\) with the norm
where
Let us estimate the norm of \(\phi _n\) in \(L^2(0,1)\):
since \(p\in W^2_1(0,1)\) and \(\Vert \psi _n\Vert _{L^2}=1\).
2 Main Results
We consider the wave equation
with initial conditions
and with Dirichlet boundary conditions
where \({\mathcal {L}}\) is defined by
where \(p\in W^2_1(0,1)\), and q is defined as in (1.3).
In our results below, concerning the initial/boundary problem (2.1)–(2.3), as the preliminary step we first carry out the analysis in the strictly regular case for summable \(q \in L^2(0,1)\). In this case, we obtain the well-posedness in the Sobolev spaces \(W^k_{\mathcal {L}}\) associated to the operator \({\mathcal {L}}\): we define the Sobolev spaces \(W^k_{\mathcal {L}}\) associated to \({\mathcal {L}}\), for any \(k \in {\mathbb {R}}\), as the space
with the norm \(\Vert f\Vert _{W^k_{{\mathcal {L}}}}:=\Vert {\mathcal {L}}^{k/2}f\Vert _{L^2}\). The global space of distributions \({\mathcal {D}}'_{\mathcal {L}}(0,1)\) is defined as follows.
The space \(C^\infty _{\mathcal {L}}(0,1):=\textrm{Dom}({\mathcal {L}}^\infty )\) is called the space of test functions for \({\mathcal {L}}\), where we define
where \(\textrm{Dom}({\mathcal {L}}^m)\) is the domain of the operator \({\mathcal {L}}^m\), in turn defined as
The Fréchet topology of \(C^\infty _{\mathcal {L}}(0,1)\) is given by the family of norms
The space of \({\mathcal {L}}\)-distributions
is the space of all linear continuous functionals on \(C^\infty _{\mathcal {L}}(0,1)\). For \(\omega \in {\mathcal {D}}'_{\mathcal {L}}(0,1)\) and \(\phi \in C^\infty _{\mathcal {L}}(0,1)\), we shall write
For any \(\psi \in C^\infty _{\mathcal {L}}(0,1)\), the functional
is an \({\mathcal {L}}\)-distribution, which gives an embedding \(\psi \in C^\infty _{\mathcal {L}}(0,1)\hookrightarrow {\mathcal {D}}'_{\mathcal {L}}(0,1)\).
We introduce the spaces \(C^j([0,T],W^k_{\mathcal {L}}(0,1))\), given by the family of norms
where \(k\in {\mathbb {R}}, \, f\in C^n([0,T],W^k_{\mathcal {L}}(0,1)).\)
Theorem 2.1
Assume that \(p'\in L^2(0,1)\), \(q=\nu '\), \(\nu \in L^\infty (0,1)\). For any \(k\in {\mathbb {R}}\), if the initial data satisfy \((u_0,\, u_1) \in W^{1+k}_{{\mathcal {L}}}\times W^k_{{\mathcal {L}}}\), then the wave equation (2.1) with the initial/boundary problem (2.2)–(2.3) has unique solution \(u\in C([0,T], W^{1+k}_{{\mathcal {L}}})\cap C^1([0,T], W^{k}_{{\mathcal {L}}})\). It satisfies the estimates
where the constants in these inequalities are independent of \(u_0\), \(u_1\), p and q.
We note that \(p'\in L^2(0,1)\) implies that \(\Vert p\Vert _{L^\infty }\le |p(0)|+\Vert p'\Vert _{L^2(0,1)}.\) Indeed, if \(p'\in L^2(0,1)\), then
Proof
Let us apply the technique of the separation of variables (see, e.g. [13]). This method involves finding a solution of a certain form. In particular, we are looking for a solution of the form
for functions T(t), X(x) to be determined. Suppose we can find a solution of (2.1) of this form. Plugging a function \(u(t,x)=T(t)X(x)\) into the wave equation, we arrive at the equation
Dividing this equation by T(t)X(x), we have
for some constant \(\lambda \). Therefore, if there exists a solution \(u(t,x) = T(t)X(x)\) of the wave equation, then T(t) and X(x) must satisfy the equations
for some constant \(\lambda \). In addition, in order for u to satisfy the boundary conditions (2.3), we need our function X to satisfy the boundary conditions (1.2). That is, we need to find a function X and a scalar \(\lambda \), such that
Equation (2.13) with the boundary conditions (2.14) has the eigenvalues of the form (1.10) with the corresponding eigenfunctions of the form (1.16) of the Sturm–Liouville operator \({\mathcal {L}}\) generated by the differential expression (1.1).
Further, we solve the left-hand side of Eq. (2.12) with respect to the independent variable t,
It is well known [13] that the solution of the equation (2.15) with the initial conditions (2.2) is
Then the solution of Eq. (2.1) is given by
For each value of n Eq. (2.16) is a solution. By the superposition principle the sum of all these solution is also a solution
Applying the initial conditions to Eq. (2.17), we have
and multiplying both sides of each equation in (2.18) by \(g(x)\psi _m(x)\), we get
Note that
Integrating over (0, 1) in (2.19), taking into account the orthonormality of \(\psi _n\) in \(L^2(0,1)\), we obtain
Further we will prove that \(u\in C^2([0,T],L^2(0,1))\). By using the Cauchy-Schwarz inequality for any fixed t, we can deduce that
By using the Parseval identity and taking into account (1.17), we get
Now, let us estimate \(\Vert g\Vert _{L^\infty }\), where
If \(p\ge 0\) at \(x\in (0,1)\), then \(\Vert g\Vert _{L^\infty }=1\). Otherwise, when we do not have \(p\ge 0\), then
According to the last expressions, we have
For the second term in (2.20), using (1.17), the properties of the eigenvalues of the operator \({\mathcal {L}}\) and the Parseval’s identity, we obtain the following estimate
Therefore
Now, let us estimate
The second term of (2.24) gives the norm of \(\Vert gu_1\Vert ^2_{L^2}\) by the Parseval identity. Since \(\lambda _n\) are eigenvalues and \(\phi _n\) are eigenfunctions of the operator \({\mathcal {L}}\), we obtain
It is known by Parseval’s identity that
Thus,
We now consider the next estimate for the derivative
where \(\phi '(x)\), taking into account (1.11), (1.15) and (1.16), is given by
By using formula (2.26) let us estimate
According (1.13) and (1.14), there exist some \(C_0>0\), such that \(C_0<\Vert {\tilde{\psi }}_n\Vert _{L^2}<\infty \), and taking into account (1.17) we get
We follow the proof of Lemma 1 in [24] to obtain
where the constant is independent of \(\nu \) and n. Then
For the second term we obtain
since \(\{\psi _n\}\) is an orthonormal basis in \(L^2\). Using the last relations we can obtain the estimate for \(\Vert \psi '_n\Vert _{L^2}\) in the following form
Using (2.27), (2.28), (2.29), (2.25) and (2.21) we obtain
Let us get next estimates by using that \(\phi ''_n(x)=p(x)\phi '_n(x)+(q(x)-\lambda _n)\phi _n(x)\), since \(\phi _n\) is a normalised eigenfunction for \({\mathcal {L}}\) with eigenvalue \(\lambda _n\). We have
Let us estimate the second term of (2.32),
Using the property of the operator \({\mathcal {L}}\) and the Parseval identity for the last expression in (2.34), we obtain
Taking into account the last expression and (2.33), (2.34) we obtain
Let us carry out the last estimate (2.11) using that \({\mathcal {L}}^ku=\lambda _n^ku\) and Parseval’s identity,
The proof of Theorem 2.1 is complete. \(\square \)
We will now express all the estimates in terms of the coefficients, to be used in the very weak well-posedness in Sect. 4.
Corollary 2.2
Assume that \(p'\in L^2(0,1)\), \(q=\nu '\), \(\nu \in L^\infty (0,1)\). If the initial data satisfy \((u_0,\, u_1) \in L^2(0,1)\times L^2(0,1)\) and \((u_0'', \, u''_1)\in L^2(0,1)\times L^2(0,1)\), then the wave equation (2.1) with the initial/boundary problems (2.2)–(2.3) has unique solution \(u\in C([0,T], L^2(0,1))\) which satisfies the estimates
where the constants in these inequalities are independent of \(u_0\), \(u_1\), p and q.
Proof
By using inequality (2.20) we obtain
In Theorem 2.1 we obtained estimates with respect to the operator \({\mathcal {L}}\), but here we want to obtain estimates with respect to the initial data \((u_0,\, u_1)\) and functions p and q. Therefore, since \(\lambda _n\ge 1\) we can use the estimate
Thus, using (2.23) and the Parseval identity in (2.39), taking into account the last relation, we obtain
By (2.24) we have
Since \(\lambda _n\) are eigenvalues of the operator \({\mathcal {L}}\), we obtain
Since \(p,\,q\in L^\infty (0,1)\) and by Parseval’s identity, we get
thus,
To obtain the results of Sect. 4, we need estimates in terms of p, q, and \((u_0,\,u_1)\). Therefore, we proceed to the next estimates. We have
where
and according to (2.22) we obtain
For \((gu_0)''\) one obtains
Given estimates (2.42), (2.43) and (2.22), for \(\Vert \partial _tu(t,\cdot )\Vert _{L^2}\) we get
Taking (2.26), (2.31), (2.40) and (2.41) into account, we make the following estimates
According to (2.42), (2.43) and (2.22) we get
Let us now get an estimate for
We have
and carrying out estimates as in (2.33) and (2.41), we obtain
Similarly, we obtain the following estimate
Using (2.42), (2.43) and (2.22), we have
The proof of Corollary 2.2 is complete. \(\square \)
3 Non-homogeneous Equation Case
In this section, we are going to give brief ideas for how to deal with the non-homogeneous wave equation with initial/boundary conditions
where operator \({\mathcal {L}}\) is defined by
Theorem 3.1
Assume that \(p'\in L^2(0,1)\), \(q=\nu '\), \(\nu \in L^\infty (0,1)\) and \(f=f(t,x)\in C^1([0,T],L^2(0,1))\). For any \(k\in {\mathbb {R}}\), if the initial data satisfy \((u_0,\, u_1) \in W^{1+k}_{\mathcal {L}}\times W^k_{\mathcal {L}}\), then the non-homogeneous wave equation with initial/boundary conditions (3.1) has unique solution \(u\in C([0,T], W^{1+k}_{\mathcal {L}})\cap C^1([0,T], W^{k}_{\mathcal {L}})\) which satisfies the estimates
where the constants in these inequalities are independent of \(u_0\), \(u_1\), p, q and f.
Proof
The substitution
brings Eq. (3.1) to the form
We can use the eigenfunctions (1.15) of the corresponding (homogeneous) eigenvalue problem (1.5)–(1.6), and look for a solution in the series form
where
We can similarly expand the source function,
Now, since we are looking for a twice differentiable function v(t, x) that satisfies the homogeneous Dirichlet boundary conditions, we can differentiate the Fourier series (3.9) term by term and using that the \(\psi _n(x)\) satisfies the equation (1.5), we obtain
We can also twice differentiate the series (3.9) with respect to t to obtain
since the Fourier coefficients of \(v_{tt}(t,x)\) are
Differentiation under the above integral is allowed since the resulting integrand is continuous.
Substituting (3.12) and (3.11) into the equation, and using (3.10), we have
and after a slight rearrangement, we get
But then, due to the completeness,
which are ordinary differential equations for the coefficients \(v_n(t)\) of the series (3.9). By the method of variation of constants we get
where
Thus, we can write a solution of Eq. (3.8) in the form
According to (3.7), we obtain the solution of the equation (3.1) in the following form
Let us estimate \(\Vert u(t,\cdot )\Vert _{L^2}\). For this we first use the estimate
For \(I_1\) by using (2.7) for the homogeneous case we have that
Now we estimate \(I_2\) in (3.14) as
Using Holder’s inequality and taking into account that \(t\in [0,T]\) we get
since \((gf)_n(t)\) is the Fourier coefficient of the function g(x)f(t, x), and by Parseval’s identity we obtain
Since
we arrive at the inequality
Thus,
and \(I_3\) in (3.14) is evaluated similarly
We finally get
Let us estimate \(\Vert \partial _tu(t,\cdot )\Vert _{L^2}\). For this we calculate \(\partial _tu(t,x)\) as follows
then we can estimate
By using (2.8) for the homogeneous case and making estimates as in (3.19), (3.20), we obtain
For (3.4) we write
Taking (2.31) into account, we have that
For \(K_2\) in (3.21) using (2.26), (2.27), (2.28) and (2.29) we obtain
where
since \(\lambda _n\ge 1,\,n=1,2,...,\) according to (3.19). So it is enough to estimate
Thus,
For \(K_3\) in (3.21) we similarly get
Taking into account the estimates for \(K_1\), \(K_2\) and \(K_3\), we obtain
We have \(\phi _n''(x)=p(x)\phi '_n(x)+(q(x)-\lambda _n)\phi _n(x)\), so that
Using (2.10) we get
Let us estimate \(E_2\) by using (3.22)–(3.23), so that we get
We similarly get
Therefore,
Let us estimate \(\Vert u(t,\cdot )\Vert ^2_{W^k_{{\mathcal {L}}}}\):
By using (2.11) we have
Using that \({\mathcal {L}}^{\frac{k}{2}}\phi _n(x)=\lambda ^{\frac{k}{2}}\phi _n(x)\), (1.17) and following as in (3.15)–(3.18), we obtain
We similarly get
Thus,
The proof of Theorem 3.1 is complete. \(\square \)
We will now express all the estimates in terms of the coefficients, to be used in the very weak well-posedness in Sect. 4.
Corollary 3.2
Assume that \(p'\in L^2(0,1)\), \(q=\nu '\), \(\nu \in L^\infty (0,1)\) and \(f(t,x)\in C^1([0,T],L^2(0,1))\). If the initial data satisfy \((u_0,\, u_1) \in L^2(0,1)\) and \((u_0'',\, u''_1)\in L^2(0,1)\), then the non-homogeneous wave equation with initial/boundary conditions (3.1) has unique solution \(u\in C([0,T], L^2(0,1))\) such that
where the constants in these inequalities are independent of \(u_0\), \(u_1\), p, q and f.
The proof of Corollary 3.2 immediately follows from Corollary 2.2 and Theorem 3.1.
4 Very Weak Solutions
In this section we will analyse the solutions for less regular potentials q and p. For this we will be using the notion of very weak solutions.
Assume that the coefficients q, p and initial data \((u_0,\, u_1)\) are the distributions on (0, 1).
Definition 4.1
(i) A net of functions \(\left( u_\varepsilon =u_\varepsilon (t,x)\right) \) is said to be \(L^2\)-moderate if there exist \(N\in {\mathbb {N}}_0\) and \(C>0\) such that
(ii) Moderateness of data: a net of functions \((u_{0,\varepsilon }=u_{0,\varepsilon }(x))\) is said to be \(H^2\)-moderate if there exist \(N\in {\mathbb {N}}_0\) and \(C>0\) such that
Definition 4.2
(i) A net of functions \(\left( \nu _\varepsilon =\nu _\varepsilon (x)\right) \) is said to be \(L^\infty _1\)-moderate if there exist \(N\in {\mathbb {N}}_0\) and \(C>0\) such that
(ii) A net of functions \((p_\varepsilon )\) is said to be \(\log \)-\(L^\infty _1\)-moderate if there exist \(N\in {\mathbb {N}}_0\) and \(C>0\) such that
Remark 4.3
We note that for the clarity of expression, we put two condition in (4.1) explicitly. However, we note that the first one follows from the second:
The same remark applies to other conditions.
Remark 4.4
We note that such assumptions are natural for distributional coefficients in the sense that regularisations of distributions are moderate. Precisely, by the structure theorems for distributions (see, e.g. [10, 12]), we know that distributions
and we see from (4.2), that a solution to an initial/boundary problem may not exist in the sense of distributions, while it may exist in the set of \(L^\infty \)-moderate functions.
To give an example, at least for \(1\le p<\infty \), let us take \(f\in L^2(0,1)\), \(f:(0,1)\rightarrow {\mathbb {C}}\). We introduce the function
then \({\tilde{f}}:{\mathbb {R}}\rightarrow {\mathbb {C}}\), and \({\tilde{f}}\in {\mathcal {E}}'({\mathbb {R}}).\)
Let \({\tilde{f}}_\varepsilon ={\tilde{f}}*\psi _\varepsilon \) be obtained as the convolution of \({\tilde{f}}\) with a Friedrich mollifier \(\psi _\varepsilon \), where
Then the regularising net \(({\tilde{f}}_\varepsilon )\) is \(L^p\)-moderate for any \(p \in [1,\infty )\), and it approximates f on (0, 1):
Now, let us introduce the notion of a very weak solution to the initial/boundary problem (2.1)–(2.3).
Definition 4.5
Let \(p, \, \nu \in {\mathcal {D}}'(0,1)\). The net \((u_\varepsilon )_{\varepsilon >0}\) is said to be a very weak solution to the initial/boundary problem (2.1)–(2.3) if there exist a \(\log \)-\(L^\infty _1\)-moderate regularisation \(p_\varepsilon \) of p, \(L^\infty _1\)-moderate regularisation \(\nu _\varepsilon \) of \(\nu \) with \(q_\varepsilon =\nu '_\varepsilon \), \(H^2\)-moderate regularisation \(u_{0,\varepsilon }\) of \(u_0,\) and \(L^2\)-moderate regularisation \(u_{1,\varepsilon }\) of \(u_1\), such that
and \((u_\varepsilon )\), \((\partial _x u_\varepsilon )\) are \(L^{2}\)-moderate.
Describing the uniqueness of the very weak solutions amounts to “measuring” the changes on involved associated nets: negligibility conditions for nets of functions/distributions read as follows:
Definition 4.6
(Negligibility) (i) Let \((u_\varepsilon )\), \(({\tilde{u}}_\varepsilon )\) be two nets in \(L^2(0,1)\). Then, the net \((u_\varepsilon -{\tilde{u}}_\varepsilon )\) is called \(L^2\)-negligible, if for every \(N\in {\mathbb {N}}\), there exists \(C>0\) such that the following condition is satisfied
for all \(\varepsilon \in (0,1]\). In the case where \(u_\varepsilon =u_\varepsilon (t,x)\) is a net depending on \(t\in [0,T]\), then the negligibility condition can be introduced as
uniformly in \(t\in [0,T]\). The constant C can depend on N but not on \(\varepsilon \).
(ii) Let \((p_\varepsilon )\), \(({\tilde{p}}_\varepsilon )\) be two nets in \(L^\infty (0,1)\). Then, the net \((p_\varepsilon -{\tilde{p}}_\varepsilon )\) is called \(L^\infty \)-negligible, if for every \(N\in {\mathbb {N}}\), there exists \(C>0\) such that the following condition is satisfied
for all \(\varepsilon \in (0,1]\).
Let us state the “\(\varepsilon \)-parameterised problems" to be considered:
and
Definition 4.7
(Uniqueness of the very weak solution) We say that initial/boundary problem (2.1)–(2.3) has a unique very weak solution, if for all \(\log \)-\(L^\infty _1\)-moderate nets \(p_\varepsilon \), \({\tilde{p}}_\varepsilon \), such that \((p_\varepsilon -{\tilde{p}}_\varepsilon )\) is \(L^\infty \)-negligible; \(L^\infty _1\)-moderate nets \(\nu _\varepsilon \), \({\tilde{\nu }}_\varepsilon \) with \(q_\varepsilon =\nu '_\varepsilon \), \({\tilde{q}}_\varepsilon ={\tilde{\nu }}'_\varepsilon \) such that \((q_\varepsilon -{\tilde{q}}_\varepsilon )\) is \(L^\infty \)-negligible; for all \(H^2\)-moderate regularisations \(u_{0,\varepsilon },\,{\tilde{u}}_{0,\varepsilon }\), such that \((u_{0,\varepsilon }-{\tilde{u}}_{0,\varepsilon })\) are \(L^2\)-negligible and for all \(L^2\)-moderate regularisations \(u_{1,\varepsilon },\,{\tilde{u}}_{1,\varepsilon }\), such that \((u_{1,\varepsilon }-{\tilde{u}}_{1,\varepsilon })\) are \(L^2\)-negligible, we have that \(u_\varepsilon -{\tilde{u}}_\varepsilon \) is \(L^2\)-negligible.
Then we have the following properties of very weak solutions.
Theorem 4.8
(Existence) Let the coefficients p, q and initial data \((u_0,\, u_1)\) be distributions in (0, 1). Then the initial/boundary problem (2.1)–(2.3) has a very weak solution.
Proof
Since the formulation of (2.1)–(2.3) in this case might be impossible in the distributional sense due to issues related to the product of distributions, we replace (2.1)–(2.3) with a regularised equation. In other words, we regularise p, \(p'\), \(\nu \), q, \(u_0\), \(u_1\), \(u'_0\) and \(u''_0\) by some corresponding sets \(p_\varepsilon \), \(p'_\varepsilon \), \(\nu _\varepsilon \), \(q_\varepsilon \), \(u_{0,\varepsilon }\), \(u_{1,\varepsilon }\), \(u'_{0,\varepsilon }\) and \(u''_{0,\varepsilon }\) of smooth functions from \(L^ \infty (0,1)\) and \(L^2(0,1)\), respectively.
Hence, \(p_\varepsilon \) is \(\log \)-\(L^\infty _1\)-moderate regularisation of the coefficient p, and \(\nu _\varepsilon \) with \(q_\varepsilon =\nu '_\varepsilon \) is \(L^\infty _1\)-moderate regularisation of \(\nu \), \(u_{0,\varepsilon }\) is \(H^2\)-moderate regularisation of \(u_0\) and \(u_{1,\varepsilon }\) is \(L^2\)-moderate regularisation of \(u_1\). So by Definition 4.1 there exist \(N\in {\mathbb {N}}_0\) and \(C_1>0\), \(C_2>0\), \(C_3>0\), \(C_4>0\), \(C_5>0\), \(C_6>0\), \(C_7>0\), \(C_8>0\) such that
Now we fix \(\varepsilon \in (0,1]\), and consider the regularised problem (4.3). Then all discussions and calculations of Theorem 2.1 are valid. Thus, by Theorem 2.1, the Eq. (4.3) has unique solution \(u_\varepsilon (t,x)\) in the space \(C^0([0,T]; H^1(0,1))\cap C^1([0,T];L^2(0,1))\).
By Corollary 2.2 there exist \(N\in {\mathbb {N}}_0\) and \(C>0\), such that
where the constants in these inequalities are independent of p, \(p'\), \(\nu \), q, \(u_0\), \(u_1\), \(u'_0\) and \(u''_0\). Hence, \((u_\varepsilon )\) is \(L^2\)-moderate, and the proof of Theorem 4.8 is complete. \(\square \)
Remark 4.9
By
we note that the net \(\partial _tu_\varepsilon \) is also \(L^2\)-moderate.
Theorem 4.10
(Uniqueness of the very weak solution) Let the coefficients p, \(q=\nu '\) and initial data \((u_0,\, u_1)\) be distributions in (0, 1). Then the very weak solution to the initial/boundary problem (2.1)–(2.3) is unique.
Proof
We denote by \(u_\varepsilon \) and \({\tilde{u}}_\varepsilon \) the families of solutions to the initial/boundary problems (4.4) and (4.5) respectively. Setting \(U_\varepsilon \) to be the difference of these nets \(U_\varepsilon :=u_\varepsilon (t,\cdot )-{\tilde{u}}_\varepsilon (t,\cdot )\), then \(U_\varepsilon \) solves
where we set \(f_\varepsilon (t,x):=({\tilde{p}}_\varepsilon (x)-p_\varepsilon (x))\partial _x{\tilde{u}}_\varepsilon (t,x) +({\tilde{q}}_\varepsilon (x)-q_\varepsilon (x)){\tilde{u}}_\varepsilon (t,x)\) for the forcing term to the non-homogeneous initial/boundary problem (4.6).
Passing to the \(L^2\)-norm of the \(U_\varepsilon \), by using (3.24) we obtain
Since
and using the initial data of (4.6), we get
for some \(N_0>0\). Taking into account the negligibility of the nets \(u_{0,\varepsilon }-{\tilde{u}}_{0,\varepsilon }\), \(u_{1,\varepsilon }-{\tilde{u}}_{1,\varepsilon }\), \(p_\varepsilon -{\tilde{p}}_\varepsilon \) and \(q_\varepsilon -{\tilde{q}}_\varepsilon \) we get
for some \(C_1>0,\,C_2>0,\,C_3>0,\,C_4>0,\,C_5>0,\,N_0,\, N_4,\,N_6\in {\mathbb {N}}\) and all \(N_1,\,N_2,\,N_3,\,N_5\in {\mathbb {N}}\), since \({\tilde{u}}_\varepsilon \) is moderate. Then, for all \(M\in {\mathbb {N}}\) we have
The last estimate holds true uniformly in t, and this completes the proof of Theorem 4.10. \(\square \)
Theorem 4.11
(Consistency) Assume that \(p'\in L^2(0,1)\), \(q=\nu '\), \(\nu \in L^\infty (0,1)\), and let \(p_\varepsilon \) be any \(\log \)-\(L^\infty _1\)-moderate regularisation of p, \(\nu _\varepsilon \) be any \(L^\infty _1\)-moderate regularisation of \(\nu \) with \(q_\varepsilon =\nu '_\varepsilon \). Let the initial data satisfy \((u_0,\, u_1) \in L^2(0,1)\times L^2(0,1)\). Let u be a very weak solution of the initial/boundary problem (2.1)–(2.3). Then for any families \(p_\varepsilon \), \(q_\varepsilon \), \(u_{0,\varepsilon }\), \(u_{1,\varepsilon }\) such that \(\Vert u_{0}-u_{0,\varepsilon }\Vert _{L^2}\rightarrow 0\), \(\Vert u_{1}-u_{1,\varepsilon }\Vert _{L^2}\rightarrow 0\), \(\Vert p-p_{\varepsilon }\Vert _{L^\infty }\rightarrow 0\) \(\Vert q-q_{\varepsilon }\Vert _{L^\infty }\rightarrow 0\) as \(\varepsilon \rightarrow 0\), any representative \((u_\varepsilon )\) of u converges as
for \(\varepsilon \rightarrow 0\) to the unique classical solution in \(C([0,T];L^2(0,1))\) of the initial/boundary problem (2.1)–(2.3) given by Theorem 2.1.
Proof
For u and for \(u_\varepsilon \), as in our assumption, we introduce an auxiliary notation \(V_\varepsilon (t, x):= u(t,x)-u_\varepsilon (t,x)\). Then the net \(V_\varepsilon \) is a solution to the initial/boundary problem
where \(f_\varepsilon (t,x)=(p_\varepsilon (x)-p(x))\partial _xu(t,x)+(q_\varepsilon (x)-q(x))u(t,x)\). Analogously to Theorem 4.10 we have that
Since
for \(\varepsilon \rightarrow 0\) and u is a very weak solution of the initial/boundary problem (2.1)–(2.3) we get
for \(\varepsilon \rightarrow 0\). This proves Theorem 4.11. \(\square \)
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The authors are supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations and by the Methusalem programme of the Ghent University Special Research Fund (BOF) (Grant number 01M01021). Michael Ruzhansky is also supported by EPSRC grants EP/R003025/2 and EP/V005529/1, and the second author by the international internship program “Bolashak” of the Republic of Kazakhstan.
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Ruzhansky, M., Yeskermessuly, A. Wave Equation for Sturm–Liouville Operator with Singular Intermediate Coefficient and Potential. Bull. Malays. Math. Sci. Soc. 46, 195 (2023). https://doi.org/10.1007/s40840-023-01587-y
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DOI: https://doi.org/10.1007/s40840-023-01587-y