Abstract
In this paper, the Cauchy problem for linear and nonlinear wave equations is studied.The equation involves an abstract operator A in a Hilbert space H and a convolution term. Here, assuming sufficient smoothness on the initial data and on coefficients, the existence, uniqueness, regularity properties, and blow-up of solutions are established in terms of fractional powers of a given sectorial operator A. We obtain the regularity properties of a wide class of wave equations by choosing a space H and an operator A that appear in the field of physics.
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The main objective of this article is to study the existence, uniqueness, regularity, and blow-up properties of the initial value problem (IVP) for the abstract wave equatıon (WE)
where \(T\in \left( 0,\right. \left. \infty \right] \), A is a linear and f(u) is a nonlinear operator in a Hilbert space H, a is a complex-valued function on \(\mathbb {R}^{n}\), and \(*\) denotes convolution. Here, \(\varphi \left( x\right) \) and \(\psi \left( x\right) \) are the given H-valued initial functions.
The qualitative behaviours of a wide class of wave equations can be found, e.g. in [2, 4,5,6,7,8,9,10, 17,18,19] and [29,30,31,32]. Wave-type equations occur in a wide variety of physical systems, such as the propagation of waves in elastic rods, hydro-dynamical processes in plasma, and in materials science, which describes spinodal decomposition and the absence of mechanical stresses (see [1, 11, 14, 19,20,21, 24]). Note that abstract hyperbolic equations were studied, e.g., in [2, 12, 22, 23].
Unlike these studies, in this paper both linear and nonlinear abstract wave equations are considered. The \(L^{p}\)-well-posedness of the Cauchy problem (1.1)–(1.2) depends crucially on the presence of the linear operator A and nonlinear function \(f\left( u\right) \). We determine the class of operator A and function f to guarantee the existence, uniqueness, regularity properties, and blow up of solutions (1.1)–(1.2) in terms of fractional powers of the operator A. By assigning a concrete space for H and an appropriate operator A we can obtain a variety of wave equations that occur in applications. As an example, we can let \(H=l_{2}\) and choose \(A_1\) as a matrix of finite or infinite dimension, i.e.,
where \(\mathbb {N}\) denotes the set of natural numbers and \(a_{ij}\) are real numbers (see, e.g., [27, §1.18] for the space \(l_{2}^{\sigma }\)). Therefore, as a corollary of our main result, we obtain the existence, uniqueness, regularity properties, and blow-up of the following IVP:
in mixed \(L^{\mathbf {p}}\left( \mathbb {R}_{T}^{n};l_{2}\right) ,\) where \( \mathbf {p=}\left( p,p,2\right) .\)
As a second example we can choose \(H=L^{2}\left( 0,1\right) \) and \( A_2 \) a degenerate differential operator in \(L^{2}\left( 0,1\right) \) with nonlocal boundary conditions
where \(u^{\left[ i\right] }=\left( y^{\gamma }\frac{d}{dy}\right) ^{i}u \) for \(0\le \gamma <\frac{1}{2}\), \(b_{1}=b_{1}\left( y\right) \) is a contınous function, \(b_{2}=b_{2}\left( y\right) \) is a bounded function in \(y\in \) \(\left[ 0,1\right] \) for a.e. \(x\in \mathbb {R}^{n}\), \(\alpha _{k}\), \( \beta _{k}\) are complex numbers, and \(W_{\gamma }^{\left[ 2\right] ,2}\left( 0,1\right) \) is a weighted Sobolev space defined by
Moreover, our main result implies the \(L^{\mathbf {p}}\left( \Omega \right) \) -regularity property of a nonlocal mixed problem for the degenerate PDE
where the mixed norm is defined as
Note that if we let \(H=\mathbb {C}\) and let the operator A be a complex-valued function, then one can obtain the previous results in the literature.
The traditional methods of the classical theory for wave equations is very limited in its ability to handle abstract wave equations. Here, a \(L^{p}\) -estimate containing fractional degrees of A is shown for the solution. Therefore, to overcome these difficulties we implement more powerful tools of abstract harmonic analysis, operator theory, interpolation of Banach spaces, and embedding theorems of Sobolev-Lions spaces.
1 Definitions and Background
In order to state our results precisely, we introduce some notation and some function spaces.
Let E be a Banach space and let \(L^{p}\left( \Omega ;E\right) \) denote the space of strongly measurable E-valued functions that are defined on a measurable subset \(\Omega \subset \mathbb {R}^{n}\) with the norm
Let \(E_{1}\) and \(E_{2}\) be two Banach spaces, and let \(\left( E_{1},E_{2}\right) _{\theta ,p}\) for \(\theta \in \left( 0,1\right) \), \(p\in \left[ 1,\infty \right] \) denote the real interpolation spaces defined by the K-method [27, §1.3.2.], Let \(E_{1}\) and \(E_{2}\) be two Banach spaces, and \(B\left( E_{1},E_{2}\right) \) denote the space of all bounded linear operators from \(E_{1}\) to \(E_{2}\). For \(E_{1}=E_{2}=E\) that space will be denoted by \(B\left( E\right) .\)
Here,
A closed linear operator A is said to be \(\psi \)-sectorial in a Banach space E with bound \(M>0\) if \(D\left( A\right) \) and \(R\left( A\right) \) are dense on E, \(N\left( A\right) =\left\{ 0\right\} \) and
for all \(\lambda \in S_{\phi }\), \(0\le \phi <\pi \), where I is the identity operator in E, and \(D\left( A\right) \) and \(R\left( A\right) \) denote the domain and range of the operator A, respectively. It is known that (see, e.g., [27, §1.15.1] ) there exist fractional powers \(A^{\theta }\) of a sectorial operator A. Let \(E\left( A^{\theta }\right) \) denote the space \(D\left( A^{\theta }\right) \) with the graphical norm
A sectorial operator \(A\left( \xi \right) \) is said to be uniformly sectorial in E for \(\xi \in \mathbb {R}^{n},\) if \(D\left( A\left( \xi \right) \right) \) is independent of \(\xi \) and the following uniform estimate
holds for all \(\lambda \in S_{\phi }.\)
A linear operator \(A=A\left( \xi \right) \) belongs to \(\sigma \left( M_{0},\omega ,E\right) \) (see [23, § 11.2]) if \( D\left( A\right) \), \(R\left( A\right) \) are dense on E, \(N\left( A\right) =\left\{ 0\right\} \), \(D\left( A\left( \xi \right) \right) \) is independent of \(\xi \in \mathbb {R}^{n}\) and for \({\hbox {Re}}\lambda >\omega \) the uniform estimate holds
Remark 1.1
It is known (see, e.g., [22, § 1.6], Theorem 6.3) that if \(A\in \sigma \left( M_{0},\omega ,E\right) \) and \(0\le \alpha <1,\) then A generates a bounded group operator \( U_{A}\left( t\right) \) satisfying
Let \(1\le p\le q<\infty \). A function \(\Psi \in L^{\infty }(\mathbb {R} ^{n}) \) is called a Fourier multiplier from \(L^{p}(\mathbb {R}^{n};E)\) to \( L^{q}(\mathbb {R}^{n};E)\) if the map P: \(u\rightarrow \mathbb {F}^{-1}\Psi (\xi )\mathbb {F}u\) for \(u\in S(\mathbb {R}^{n};E)\) is well defined and extends to a bounded linear operator
Let E be a Banach space and let \(S=S(\mathbb {R}^{n};E)\) denote E-valued Schwartz class, i.e., the space of all E-valued rapidly decreasing smooth functions on \(\mathbb {R}^{n}\) equipped with its usual topology generated by seminorms. Let \(S(\mathbb {R}^{n};\mathbb {C})\) be denoted by S. Let \( S^{\prime }(\mathbb {R}^{n};E)\) denote the space of all continuous linear functions from S into E equipped with the bounded convergence topology. Recall that \(S(\mathbb {R}^{n};E)\) is norm dense in \(L^{p}(\mathbb {R}^{n};E)\) when \(1\le p<\infty \).
The Fourier transformation of the operator function \(B\left( x\right) \) with domain \(D\left( B\right) \) independent on \(x\in \mathbb {R}^{n}\) is a generalized function defined as
(For details see, e.g., [2, Section 3].)
Definition 1.1
Let U be an open set in a Banach space X, and let Y be a Banach space. A function f : \(U\rightarrow Y\) is called (Fr échet) differentiable at x \(\in U\) if there is a bounded linear operator Df(x) : \(X\rightarrow Y\), called the derivative of f at a, such that
If f is differentiable at each \(x\in U\), then f is said to be differentiable. This function may also have a derivative, the second-order derivative of f, which, by the definition of derivative, will be a map
Let m be a positive integer, and let \(W^{m,p}\left( \Omega ;E\right) \) denote an \(E-\)valued Sobolev space of all functions \(u\in L^{p}\left( \Omega ;E\right) \) that have the generalized derivatives \(\frac{\partial ^{m}u}{ \partial x_{k}^{m}}\in L^{p}\left( \Omega ;E\right) \) with the norm
Let \(W^{s,p}\left( \mathbb {R}^{n};E\right) \) denote the fractional Sobolev space of order \(s\in \mathbb {R}\), defined as
It is clear that \(W^{0,p}\left( \mathbb {R}^{n};E\right) =L^{p}\left( \mathbb { R}^{n};E\right) \). Let \(E_{0}\) and E be two Banach spaces and let \(E_{0}\) be continuously and densely embedded into E. Here, \(W^{s,p}\left( \mathbb {R }^{n};E_{0},E\right) \) denotes a Sobolev- Lions- type space, i.e.,
In a similar way, we define the following Sobolev- Lions- type space:
Let \(L_{q}^{*}\left( E\right) \) denote the space of all E-valued function space such that
Let \(s>0\). The Fourier-analytic representation of an E-valued Besov space on \(\mathbb {R}^{n}\) is defined as
It should be note that the norm of a Besov space does not depend on \( \varkappa \) (see, e.g., [27, § 2.3] for the case \(E= \mathbb {C}\)).
Let
Definition 1.1
For all \(T>0,\) the function \(u\in C^{2}\left( \left[ 0,T\right] ;Y_{\infty }^{2,s,p}\left( A,H\right) \right) \) that satisfies the equation \((1.1)--(1.2)\) a.e. in \(\mathbb {R}_{T}^{n}\) is called the continuous solution or the strong solution of the problem \((1.1)--(1.2)\). If \(T<\infty \), then \(u\left( x,t\right) \) is called the local strong solution of the problem \((1.1)-(1.2)\). If \(T=\infty \), then \(u\left( x,t\right) \) is called the global strong solution of (1.1)–(1.2).
Sometimes we use one and the same symbol C without distinction to denote various positive constants that may differ from each other even in a single context. When we want to specify the dependence of such a constant on a parameter, say \(\alpha \), we write \(C_{\alpha }\). Moreover, for u, \( \upsilon ,>0\) the relations \(u\lesssim \upsilon \), u \(\approx \) \(\upsilon \) mean that there exist positive constants C, \(C_{1},\) \(C_{2}\) independent of u and \(\upsilon \) such that, respectively,
The paper is organized as follows: In Sect. 2, some definitions and background are given. In Sect. 2, we obtain the existence of a unique solution and a priori estimates for the solution of the linearized problem (1.1)–(1.2). In Sect. 3, we show the existence and uniqueness of a local strong solution of the problem (1.1)–(1.2). In Sect. 4, the existence and uniqueness of a global strong solution of the problem (1.1)–(1.2) is derived. Section 5 is devoted to the blow-up property of the solution of (1.1)–(1.2) . In Sect. , we show some applications of the problem (1.1)–(1.2).
2 Estimates for the Linearized Equation
In this section, we make necessary estimates for solutions of the Cauchy problem for the nonlocal linear WE
where A is a linear operator in a Hilbert space H and a is a complex-valued function on \(\mathbb {R}^{n}.\) Let
where \(\left( Y^{s,p},X_{p}\right) _{\theta ,p}\) denotes the real interpolation space between \(Y^{s,p}\) and \(X_{p}\) for \(\theta \in \left( 0,1\right) \), \(p\in \left[ 1,\infty \right] \) (see, e.g., [27, §1.3]).
Remark 2.1
By properties of real interpolation of Banach spaces and interpolation of the intersection of spaces (see, e.g., [27, §1.3]), we obtain
In a similar way, we have
Remark 2.2
Let A be a densely defined operator on a Banach space. Let A be a sectorial operator in a Hilbert space H. In view of interpolation by the domain of sectorial operators (see, e.g., [27, §1.8.2]) we have the following relation:
for \(0<\theta < 1\) and \(0<\varepsilon < 1-\theta \).
Note that from the result of J. Lions - J.Peetre result (see, e.g., [27, §1.8.2], we obtain the following result:
Lemma A\(_{1}\). The trace operator \(u\rightarrow \frac{\partial ^{i}u }{\partial t^{i}}\left( x,t\right) \) is bounded and continuous from \( Y^{2,s,p}\left( A,H\right) \) onto
Let
Let A be a generator of the strongly continuous cosine operator-function in H defined by formula
(see, e.g., [3, §3, 12, §11]). Then, from the definition of the sine operator-function \(S\left( t\right) ,\) we have
Remark 2.3
Let A be a densely defined operator in a Hilbert space H. By virtue of [3, Theorem 3.15.3.] , if A is a generator of a cosine function \(C\left( t\right) \), i.e.,
then there exist \(\omega \), \(M\ge 0\) such that \(A\in \sigma \left( M_{0},\omega ,H\right) \).
Condition 2.1
Assume the following: (1) there exists \(\hat{a}\in C^{\left( m\right) }\left( \mathbb {R}^{n}\right) \) such that
(2) A is \(\psi \)-sectorial in H for \(\psi<\) \(\pi -\psi _{1}\) and A is a generator of the cosine function; (3) \(A_{\xi }\ne 0\) for all \(\xi \in \mathbb {R}^{n}\).
In view of Condition 2.1 and by virtue of [3, § 3] (or [12, §11]) we obtain that \(A_{\xi }\) is a generator of the strongly continuous cosine and sine operator function defined by
First we need the following lemmas:
Lemma 2.1
Let Condition 2.1 hold. Then problem (2.1)–(2.2) has a strong solution.
Proof
Using the Fourier transform, we get from (2.1)-(2.2)
where \(\hat{u}\left( \xi ,t\right) \) is the Fourier transform of \(u\left( x,t\right) \) in x and \(\hat{\varphi }\left( \xi \right) \), \(\hat{\psi } \left( \xi \right) \) are the Fourier transforms of \(\varphi \) and \(\psi \), respectively. By virtue of [3, § 3, 12, § 11] we obtain that \(A_{\xi }\) is a generator of a strongly continuous cosine operator function and that problem (2.4) has a unique solution for all \( \xi \in \mathbb {R}^{n}\) that can be expressed as
for all \(\xi \in \mathbb {R}^{n}\), i.e., problem (2.1)–(2.2) has a unique solution
where \(C_{1}\left( t\right) \), \(S_{1}\left( t\right) \), Q are linear operator functions defined by
Now, we can show the main results of this section.
Theorem 2.1
Assume that Condition 2.1 holds and
for \(p\in \left[ 1,\infty \right] \) and some \(q\in \left[ 1,2\right] \). Let \( 0\le \alpha <1-\frac{1}{2p}\). Then for \(\varphi \in X_{1}\left( A^{\alpha }\right) \mathbb {\cap }\) \(\mathbb {H}_{0p}\), \(\psi \in X_{1}\left( A^{\alpha }\right) \cap \) \(\mathbb {H}_{1p}\), \(g\left( .,t\right) \in Y_{1}^{s,p},t\in \left[ 0,T\right] ,\) and \(g\left( x,.\right) \in L^{1}\left( 0,T;Y_{1}^{s,p}\right) \), \(x\in \mathbb {R}^{n},\) problem \((2.1)-(2.2)\) has a unique solution \(u(x,t)\in C^{2}\left( \left[ 0,T\right] ;X_{\infty }\right) ,\) and the following uniform estimate holds:
moreover, for \(\varphi \in X_{1}\left( A^{\frac{1}{2}+\alpha }\right) \mathbb {\cap }\) \(\mathbb {H}_{0p}\), \(\psi \in X_{1}\left( A^{\frac{1}{2} +\alpha }\right) \cap \) \(\mathbb {H}_{1p}\) and \(g\left( .,t\right) \in Y_{1}^{s,p}\left( A^{\frac{1}{2}}\right) \) we have
uniformly in \(t\in \left[ 0,T\right] \), where the constant \(C_{0}>0\) depends only on A, H, and initial data.
Proof
By Lemma 2.1, the problem (2.1)–(2.2) has a solution \( u(x,t)\in C^{2}\left( \left[ 0,T\right] ;\right. \left. Y^{s,p}\left( A;H\right) \right) \) for \(\varphi \in X_{1}\left( A^{\alpha }\right) \), \(\psi \in \mathbb {H}_{1p}\) , \(g\left( .,t\right) \in Y_{1}^{s,p},\) and \(g\left( x,.\right) \in L^{1}\left( 0,T;Y_{1}^{s,p}\right) \). Let \(N\in \mathbb {N}\) and
From (2.6) we deduce that
By virtue of Remarks 2.1, 2.2 and the properties of sectorial operators, we get the following uniform estimate
Hence, due to the uniform boundedness of operator functions \(C\left( \xi ,t\right) \), \(S\left( \xi ,t\right) \), in view of \(\left( 2.3\right) ,\) and by Minkowski’s inequality for integrals, we get the uniform estimate
Let
Moreover, from (2.6) we deduce that
where here the space \(L^{\infty }\left( \Omega ;H\right) \) is denoted by \( L^{\infty }\). It is clear that
where
Using the resolvent properties of sectorial operators, we have
Then by calculating \(\frac{\partial }{\partial \xi _{k}}\Phi _{0}\left( \xi \right) \), \(\frac{\partial }{\partial \xi _{k}}\Phi _{1}\left( \xi \right) \) , we obtain
Let us show that \(G_{i}\left( .,t\right) \in B_{q,1}^{n\left( \frac{1}{q}+\frac{1}{p}\right) }\left( \mathbb {R}^{n};B\left( H\right) \right) \) for some \(q\in \left( 1,2\right) \) and for all \(t\in \left[ 0,T \right] \), where
By the embedding properties of Sobolev and Besov spaces it sufficient to derive that \(G_{i}\in W_{q}^{\sigma }\left( \mathbb {R}^{n};B\left( H\right) \right) \) for some \(\sigma \) \(>n\left( \frac{1}{q}+\frac{1}{p}\right) \). Indeed, by contraction, Condition 2.1, and by (2.12) we get \( G_{i}\in L^{q}\left( \mathbb {R}^{n};B\left( H\right) \right) \). For deriving the embedding relations \(G_{i}\in W_{q}^{\sigma }\left( \mathbb {R} ^{n};B\left( H\right) \right) \), it suffices to show that
Indeed, in view of (2.12), \(\left( 1+\left| \xi \right| ^{2}\right) ^{\frac{\sigma }{2}}\Phi _{i}\left( \xi \right) \) are uniformly bounded for \(\xi \in \mathbb {R}^{n}\). By virtue of (2.3), (2.13), by assumption \(\left( 2.7\right) ,\) and in view of Remark 2.3, we have
for
Hence by the Fourier multiplier theorems (see, e.g., [13, Theorem 4.3]) we get that the functions \(G_{i}\left( \xi ,t\right) \) are Fourier multipliers from \(L^{p}\left( \mathbb {R}^{n};H\right) \) to \( L^{\infty }\left( \mathbb {R}^{n};H\right) \). In a similar way we obtain that
are \(L^{p}\left( \mathbb {R}^{n};H\right) \rightarrow L^{\infty }\left( \mathbb {R}^{n};H\right) \) Fourier multipliers. Then by Minkowski’s inequality for integrals, from (2.3), (2.10) -(2.12) and by Remark 2.3 we have
Moreover, by virtue of Remark 2.1–2.3 and by reasoning as above, we have the following estimate:
uniformly in \(t\in \left[ 0,T\right] \). Thus, from \(\left( 2.6\right) \), \( \left( 2.14\right) ,\) and (2.15) we obtain
By differentiating (2.6) in a similar way, we get the second inequality
Then from (2.16) and (2.17), in view of Remarks 2.1, 2.2, we obtain the estimate (2.8).
Let us now show that problem (2.1)–(2.2) has a unique solution \(u\in C^{\left( 1\right) }\left( \left[ 0,T\right] ;Y^{s,p}\right) \) . Suppose that the problem \((2.1)-\left( 2.2\right) \) has two solutions \( u_{1},\) \(u_{2}\in C^{\left( 1\right) }\left( \left[ 0,T\right] ;Y^{s,p}\right) \). Then by the linearity of (2.1), we get that \(\upsilon =u_{1}-u_{2}\) is also a solution of the corresponding homogenous equation
Moreover, by (2.16) we have the following estimate
Since \(N\left( A\right) =\left\{ 0\right\} \), the above estimate implies that \(\upsilon =0\), i.e., \(u_{1}=u_{2}.\)
Theorem 2.2
Assume that Condition 2.1 and \(\left( 2.7\right) \) are satisfied. Let \(0\le \alpha <1-\frac{1}{2p}\). Then for \(\varphi \), \(\psi \in Y^{s,p}\left( A^{\alpha }\right) \), \(g\left( .,t\right) \in Y^{s,p}\), \( t\in \left[ 0,T\right] ,\) and \(g\left( .,t\right) \in L^{1}\left( 0,T;Y^{s,p}\right) \), \(x\in \mathbb {R}^{n},\) problem (2.1)–(2.2) has a unique solution \(u\in C^{2}\left( \left[ 0,T \right] ;Y^{s,p}\right) \) and the following uniform estimate holds:
Moreover, for \(\varphi \), \(\psi \in Y^{s,p}\left( A^{\alpha +\frac{ 1}{2}}\right) \) and \(g\left( x,t\right) \in Y^{s,p}\left( A^{\frac{1}{2} }\right) \) we have the following estimate
for all \(t\in \left[ 0,T\right] .\)
Proof
From (2.5) and (2.1) we get the following uniform estimate:
Using the Fourier multiplier theorem [13, Theorem 4.3] and reasoning as in Theorem 2.1 we get that \(\left( 1+\left| \xi \right| ^{2}\right) ^{-\frac{s}{2}}C\left( \xi ,t\right) \), \(\left( 1+\left| \xi \right| ^{2}\right) ^{-\frac{s}{2}}S\left( \xi ,t\right) \) and \(\left( 1+\left| \xi \right| ^{2}\right) ^{-\frac{s}{ 2}}A^{\alpha }S\left( \xi ,t\right) \) are Fourier multipliers in \( L^{p}\left( \mathbb {R}^{n};H\right) \) uniformly with respect to \(t\in \left[ 0,T\right] \). So, the estimate (2.19) by using the Minkowski’s inequality for integrals implies \(\left( 2.18\right) .\)
The uniqueness of (2.1)–(2.2) is obtained by reasoning as in Theorem 2.1.
3 Local Well Posedness of IVP for Nonlinear WE
In this section, we will show the local existence and uniqueness of a solution of the nonlinear problem (1.1)–(1.2).
For this we need the following lemmas. Reasoning as in [5, 15, 31], we prove the following lemmas concerning the behaviour of the nonlinear term in the E-valued space \(Y^{s,p}\). Here, let H be a Banach algebra.
Lemma 3.1
Let \(s\ge 0\), \(f\in C^{\left[ s\right] +1}\left( \mathbb {H};H\right) \) with \(f(0)=0\). Then for all \(u\in Y^{s,p}\cap L^{\infty }\), we have \(f(u)\left( .\right) \in Y^{s,p}\cap X_{\infty }\). Moreover, there is a constant A(M) depending on M such that for all \( u\in Y^{s,p}\cap L^{\infty }\) with \(\left\| u\right\| _{X_{\infty }}\le M,\)
Using Lemma 3.1 and properties of convolution operators we obtain the following corollary:
Corollary 3.1
Let \(s\ge 0\), \(f\in C^{\left[ s\right] +1}\left( \mathbb {R};H\right) \) with \(f(0)=0\). Moreover, assume \(\Phi \in L^{\infty }\left( \mathbb {R}^{n};B\left( H\right) \right) \). Then for all \(u\in Y^{s,p}\cap L^{\infty }\) we have, \(f(u)\in Y^{s,p}\cap X_{\infty }\). Moreover, there is a constant A(M) depending on M such that for all \( u\in Y^{s,p}\cap L^{\infty }\) with \(\left\| u\right\| _{X_{\infty }}\le M,\)
Lemma 3.2
Let \(s\ge 0\), \(f\in C^{\left[ s\right] +1}\left( \mathbb {R};H\right) \). Then for every M there is a constant K(M) depending on M such that for all u, \(\upsilon \in Y^{s,p}\cap X_{\infty } \) with \(\left\| u\right\| _{X_{\infty }}\le M\), \(\left\| \upsilon \right\| _{X_{\infty }}\le M\), \(\left\| u\right\| _{Y^{s,p}}\le M\), \(\left\| \upsilon \right\| _{Y^{s,p}}\le M,\)
Reasoning as in [30, Lemma 3.4] and [15, Lemma X 4] we have, respectively:
Corollary 3.2
Let \(s>\frac{n}{2}\), \(f\in C^{\left[ s\right] +1}\left( \mathbb {R};H\right) \). Then for every positive M there is a constant K(M) depending on M such that for all u, \(\upsilon \in Y^{s,p} \) with \(\left\| u\right\| _{Y^{s,p}}\le M\), \(\left\| \upsilon \right\| _{Y^{s,p}}\le M,\)
Lemma 3.3
If \(s>0\), then \(Y_{\infty }^{s,p}\) is an algebra. Moreover, for f, \(g\in Y_{\infty }^{s,p},\)
Using Corollary 3.1 and Lemma 3.3 we obtain the following results:
Lemma 3.4
Let \(s\ge 0\), \(f\in C^{\left[ s\right] +1}\left( \mathbb {R};H\right) ,\) and \(f\left( u\right) =O\left( \left| u\right| ^{\gamma +1}\right) \) for \(u\rightarrow 0\), \(\gamma \ge 1\) a positive integer. If \(u\in Y_{\infty }^{s,p}\) and \(\left\| u\right\| _{X_{\infty }}\le M\), then
Corollary 3.3
Let \(s\ge 0\), \(f\in C^{\left[ s\right] +1}\left( \mathbb {R};H\right) ,\) and \(f\left( u\right) =O\left( \left| u\right| ^{\gamma +1}\right) \) for \(u\rightarrow 0\), \(\gamma \ge 1\) a positive integer. Moreover, assume \(\Phi \in L^{\infty }\left( \mathbb {R} ^{n};B\left( H\right) \right) \). If \(u\in Y_{\infty }^{s,p}\) and \(\left\| u\right\| _{X_{\infty }}\le M\), then
Lemma 3.5
Let \(s\ge 0\), \(f\in C^{\left[ s\right] +1}\left( \mathbb {R};H\right) ,\) and \(f\left( u\right) =O\left( \left| u\right| ^{\gamma +1}\right) \) for \(u\rightarrow 0\). Moreover, let \( \gamma \ge 0\) be a positive integer. If u, \(\upsilon \in Y_{\infty }^{s,p} \), \(\left\| u\right\| _{Y^{s,p}}\le M\), \(\left\| \upsilon \right\| _{Y^{s,p}}\le M\) and \(\left\| u\right\| _{X_{\infty }}\le M\), \(\left\| \upsilon \right\| _{X_{\infty }}\le M\), then
Let \(\mathbb {H}_{0}\) denote the real interpolation space between \( Y^{s,p}\left( A,H\right) \) and \(X_{p}\) with \(\theta =\frac{1}{2p}\), i.e.,
Remark 3.0
Let \(u\in Y^{2,s,p}=W^{2,s,p}\left( \mathbb {R} _{T}^{n};H\left( A\right) ,H\right) \). Then by a result of J. Lions and J. Peetre (see e.g. [27, §1.8.2] the trace operator \( u\rightarrow \frac{\partial ^{i}u}{\partial t^{i}}\left( x,t\right) \) is bounded from \(Y^{2,s,p}\) to\(\ C\left( \mathbb {R}^{n};\left( Y^{s,p},X_{p}\right) _{\theta _{j},p}\right) \), where
Moreover, if \(u\left( x,.\right) \in \left( Y^{s,p},X_{p}\right) _{\theta _{j},p}\), then under some assumptions that will be stated in Sect. 3, \( f\left( u\right) \in H\) for all x, \(t\in \mathbb {R}_{T}^{n}\) and the map \( u\rightarrow f\left( u\right) \) is bounded from \(\left( Y^{s,p},X_{p},\right) _{\frac{1}{2p},p}\) into H. Hence, the nonlinear equation (1.1) is satisfied in the Banach space H. Here, \( H\left( A\right) \) denotes a domain of A equipped with the graphical norm, \(\left( Y^{s,p},X_{p}\right) _{\theta ,p}\) is a real interpolation space between \(Y^{s,p}\) and \(X_{p}\) for \(\theta \in \left( 0,1\right) \), \(p\in \left[ 1,\infty \right] \) (see, e.g., \(\left[ 27,\ \S 1.3\right] \)).
Remark 3.1
Using a result of J. Lions-I. Petree (see, e.g., [27, § 1.8]) we obtain that the map \(u\rightarrow u\left( t_{0}\right) \), \(t_{0}\in \left[ 0,T\right] \) is continuous and surjective from \(Y^{2,s,p}\left( A,H\right) \) onto \(\mathbb {H}_{0p}\) and there is a constant \(C_{1}\) such that
Let
Condition 3.1
Assume the following:
-
(1)
Condition 2.1 holds, \(0\le \alpha <1-\frac{1}{2p}\) and
$$\begin{aligned} s>\frac{2p}{2p-1}\left( \frac{2n}{q}+\frac{1}{p}\right) , q\in \left[ 1,2\right] , p\in \left[ 1,\infty \right] ; \end{aligned}$$ -
(2)
the function \(u\rightarrow f\left( u\right) \) is continuous from \(u\in \mathbb {H}_{0p}\) into E, \(f\in C^{k}\left( \mathbb {H};H\right) \) with k an integer, \(k\ge s>\frac{n}{p},\) and \(f\left( u\right) =O\left( \left| u\right| ^{\gamma +1}\right) \) for \(u\rightarrow 0\), where \(\gamma \ge 1 \) is a positive integer.
Let
The main aim of this section is to prove the following results:
Theorem 3.1
Let Condition 3.1 hold. Then there exists a constant \( \delta >0\) such that for all \(\varphi \in \) \(Y_{0}\left( A^{\alpha }\right) \) and \(\psi \in \) \(Y_{1}\left( A^{\alpha }\right) \) satisfying
problem (1.1)–(1.2) has a unique local strong solution \(u\in C^{2}\left( Y_{1}^{s,p}\left( A\right) \right) \). Moreover,
where the constant C depends only on A, H, g, f, and initial values.
Proof
By (2.5), (2.6), the problem of finding a solution u of (1.1)–(1.2) is equivalent to finding a fixed point of the mapping
where \(C_{1}\left( t\right) \), \(S_{1}\left( t\right) \) are defined by (2.6) and \(Q\left( u\right) \) is the map defined by
We define the metric space
equipped with the norm defined by
where \(\delta >0\) satisfies (3.7) and \(C_{0}\) is the constant in Theorems 2.1 and 2.2. It is easy to prove that \(C\left( T,A\right) \) is a complete metric space. From embedding of the Sobolev–Lions space \( Y^{s,p}\left( A,H\right) \) (see, e.g., \(\left[ \text {30}\right] \), Theorem 1) and the trace result \(\left( 3.6\right) \) we obtain that \(\left\| u\right\| _{X_{\infty }}\le 1\) if we take \(\delta \) small enough. For \( \varphi \in \) \(Y_{0}\left( A^{\alpha }\right) \) and \(\psi \in \) \(Y_{1}\left( A^{\alpha }\right) \), let
We will find T and M that G is a contraction in \(C^{2,s,p}\left( T,A\right) \). By Theorems 2.1, 2.2 and Corollary 3.3\(f\left( u\right) \in Y_{1}^{s,p}\). So, problem (1.1)–(1.2) has a solution satisfying
where \(C_{1}\left( t\right) \), \(S_{1}\left( t\right) \) are defined by (2.5) and (2.6). By our assumptions, it is easy to see that the map G is well defined for \(f\in C^{\left[ s\right] +1}\left( \mathbb {H}_{0p};H\right) \). First, let us prove that the map G has a unique fixed point in \(C\left( T,A\right) \). For this, it is sufficient to show that the operator G maps \(C\left( T,A\right) \) into \( C\left( T,A\right) \) and G is strictly contractive if \(\delta \) is sufficiently small. In fact, by (2.7) in Theorem 2.1, Corollary 3.3, and in view of (3.7), we have
On the other hand, by (2.17), Corollary 3.3, and (3.7), we get
Hence combining (3.11) with (3.12), we obtain
So taking \(\delta \) small enough that \(C\left( 5C_{8}\delta \right) ^{\gamma }<\frac{1}{5}\), by Theorems 2.1, 2.2 and (3.13), G maps \( C\left( T,A\right) \) into \(C\left( T,A\right) \).
Now we are going to prove that the map G is strictly contractive. Let \( u_{1}\), \(u_{2}\in \) \(C\left( T,A\right) \) be given. From (3.10) we get
By (2.7) in Theorem 2.1 and Corollary 3.3, we have
On the other hand, by (2.17) in Theorem 2.2, Corollary 3.3, and (3.7), we get
Combining (3.14) with (3.15) yields
Taking \(\delta \) small enough, from (3.16) we obtain that G is strictly contractive in \(C\left( T,A\right) \). Using the contraction mapping principle, we get that \(G\left( u\right) \) has a unique fixed point \( u\left( x,t\right) \in C\left( T,A\right) \) and \(u\left( x,t\right) \) is the solution of (1.1)–(1.2).
Let us show that this solution is unique in \(C^{2,s}\left( A,H\right) \). Let \(u_{1}\), \(u_{2}\in C^{2,s}\left( A,H\right) \) be two solutions of (1.1)–(1.2). Then for \(u=u_{1}-u_{2}\), we have
Hence by Minkowski’s inequality for integrals and by Theorem 2.2, from \( \left( 3.17\right) \) we obtain
From (3.18) and Gronwall’s inequality, we have \(\left\| u_{1}-u_{2}\right\| _{Y^{s,p}}=0\), i.e., problem (1.1)–(1.2) has a unique solution in \(C^{2,s}\left( A,H\right) \).
Consider the problem (1.1)–(1.2) when \(\varphi \in \mathbb {H}_{0p}\) and \(\psi \in \mathbb {H}_{1p}\). Let
Condition 3.2
Assume the following: (1) Condition 2.1 holds; (2) \( 0\le \alpha <1-\frac{1}{2p}\), \(\varphi \in \) \(\mathbb {H}_{0p}\), \(\psi \in \) \(\mathbb {H}_{1p},\) and
(3) \(f\in C^{\left[ s\right] +1}\left( \mathbb {H};H\right) \) with \(f(0)=0\).
Reasoning as in Theorem 3.1 and [13, Theorem 1.1] , we have the following:
Theorem 3.2
Let Condition 3.2 hold. Then there exists a constant \(\delta >0\) such that for all \(\varphi \in \) \(\mathbb {H}_{0p}\), \(\psi \in \) \( \mathbb {H}_{1p}\) satisfying
problem (1.1)–(1.2) has a unique local strong solution \(u\in C^{\left( 2\right) }\left( Y^{s,p}\right) \). Moreover,
where the constant C depends only on f and initial data.
Proof
Consider the metric space defined by
equipped with the norm
where \(\delta >0\) satisfies (3.19) and \(C_{0}\) is the constant in Theorem 2.1. It is easy to prove that \(W_{0}^{s,p}\) is a complete metric space. From Sobolev embedding theorem we know that \( \left\| u\right\| _{\infty }\le 1\) if we take \(\delta \) small enough. By Theorem 2.2 and Corollary 3.1, \(f\left( u\right) \in Y^{s,p}\). Thus problem (1.1) –(1.2) has a unique solution, which can be written as (3.9). We should prove that the operator \( G\left( u\right) \) defined by (3.9) is strictly contractive if \(\delta \) is sufficiently small. In fact, by (2.17) in Theorem 2.2 and Lemma 3.1, we get
where
Therefore, from (3.20) we have
Taking \(\delta \) small enough that \(C\left( 3C_{0}\delta \right) ^{\alpha }\) \(<1/3\), from (3.22) and from Theorems 2.1, 2.2 we get that G maps \(W_{0}^{s,p}\) into \(W_{0}^{s,p}\). Then reasoning as in Theorem 3.1, we obtain that G : \(W_{0}^{s,p}\rightarrow W_{0}^{s,p}\) is strictly contractive. Using the contraction mapping principle, we know that G(u) has a unique fixed point \(u\in \) \(C^{\left( 2\right) }\left( Y^{s,2}\right) \) and u(x, t) is the solution of problem \((1.1)-(1.2)\). Moreover, by virtue of Theorem 2.1, from (3.19) we obtain (3.20) .
We claim that the solution of (1.1)-(1.2) is also unique in \(C^{\left( 1\right) }\left( Y^{s,2}\right) \). In fact, let \(u_{1}\) and \(u_{2}\) be two solutions of the problem (1.1)–(1.2) and \(u_{1}\), \(u_{2}\in C^{\left( 2\right) }\left( Y^{s,2}\right) \). Using the contraction mapping principle, we know that G(u) has a unique fixed point \(u\in \) \(C^{\left( 2\right) }\left( Y^{s,2}\right) \). Let \(u=u_{1}-u_{2}\). Then
This fact is derived in a similar way to what was done in Theorem 3.1, using Theorems 2.1, 2.2 and Gronwall’s inequality.
Let
Theorem 3.3
Let Condition 3.2 hold. Then there exist \(T>0\) such that problem (1.1)–(1.2) is well posed with solution in \(C^{1}\left( \left[ 0,T\right] ;Y^{s,p}\left( A,H\right) \right) \) for initial data \(\varphi \in \) \(\mathbb {H}_{0p}\) and \(\psi \in \) \(\mathbb {H}_{1p}.\)
Proof
Consider the operator \(u\rightarrow f\left( u\right) \). By Corollary 3.1, \(f\left( u\right) \) is locally Lipschitz on \(Y^{s,p}\). Then reasoning as in Theorem 3.2 and [13, Theorem 1.1], we obtain that G: \(W_{0}^{s,p}\rightarrow W_{0}^{s,p}\) is strictly contractive. Using the contraction mapping principle, we get that the operator G(u) defined by (3.5) has a unique fixed point \( u(x,t)\in \) \(C^{\left( 2\right) }\left( Y^{s,p}\right) \) and u(x, t) is the solution of the problem \((1.1)-(1.2)\). Moreover, we show that the solution u(x, t) of \((1.1)-(1.2)\) is also unique in \(C^{\left( 2\right) }\left( Y^{s,p}\right) \). In fact, let \(u_{1}\) and \(u_{2}\) be two solutions of the problem \((1.1)-(1.2)\) and \(u_{1}\), \(u_{2}\in C^{\left( 2\right) }\left( Y^{s,p}\right) \). Let \(u=u_{1}-u_{2}\). Then
This fact is derived in a similar way to what was done in Theorem 3.2, by using Theorems 2.1, 2.2 and Gronwall’s inequality.
The solution in Theorems 3.2–3.3 can be extended to a maximal interval \( [0,T_{\max })\), where finite \(T_{\max }\) is characterized by the blow-up condition
Lemma 3.8
Let Condition 3.2 hold and u be a solution of \((1.1)-(1.2)\). Then there is a global solution if for all \(T<\infty \), we have
Proof
Indeed, reasoning as in the second part of the proof of Theorem 3.1, using a continuation of the local solution of \((1.1)-(1.2),\) and assuming contrary that (3.24) holds and \(T_{0}<\infty ,\) we obtain a contradiction, i.e., we get \(T_{0}=T_{\max }=\infty .\)
4 Conservation of Energy and Global Existence
In this section, we prove the existence and uniqueness of the global strong solution for the problem \((1.1)-(1.2).\) For this purpose, we are going to make a priori estimates of the strong solution of \((1.1)-(1.2)\). Here, the scalar product of u, \(\upsilon \in X_{2}\) will be denoted just by \(\left( u,\upsilon \right) \). Moreover, the norm of \(u\in X_{2}\) will be denoted by \( \left\| u\right\| \).
Let
where \(Y^{s,p}\left( A;H\right) \) is as defined in Sect. 2.
First, we prove the following
lemma:
Lemma 4.1
Let Condition 3.2 hold and \(0\le \alpha <1-\frac{1}{2p}\) . Assume there exist a solution \(u\in C^{\left( 2,s\right) }\left( A,H\right) \) of (1.1)–(1.2). Then
Proof
By Lemma 2.1, problem (1.1)–(1.2) is equivalent to the integral equation,
where \(C_{1}\left( t\right) \), \(S_{1}\left( t\right) \) are operator functions defined by (2.5) and (2.6), g replaced by \(f\left( u\right) ,\) and
From (4.1) we get that
Since \(C_{1}\left( t\right) ,\) \(S_{1}\left( t\right) ,\) and \(\frac{d}{dt} S\left( \xi ,t\right) \) are uniformly bounded operators in E for fixed t, by (4.2) and Fourier multiplier results in \(X^{p}\) spaces (see, e.g., [12, Theorem 4.3]), we have
By differentiating (2.3), in a similar way we have
For fixed t, we have \(f(u)\in Y^{s,p}\). Moreover, by the assumption on A we have the uniform estimate
Due to \(s+r\ge 1,\) from (4.2) and Fourier multiplier results in \(X_{p}\) we get
Then from (4.1) and (4.3)–(4.6) we obtain the assertion.
Lemma 4.2
Assume Condition 3.2 holds with \(a=0\). Suppose a solution of \((1.1)-(1.2)\) exists in \(C^{(2,s)}(A,H) \). If \(\psi \in X^{p},\) then \(u_{t}\in C^{(1) }( X^{p}) \). Moreover, if \(\varphi \in X^{p}\), then \(u\in C^{(1)}(X^{p}).\)
Proof
Integrating equation (1.1) for \(a=0\) twice and calculating the resulting double integral as an iterated integral, we have
From (4.8), for fixed t and \(\tau \) we get \(f\left( u\right) \in Y^{s,p}\) for all t. Also
Then from (4.7)–(4.9) we obtain
By the assumption on A, g and for fixed \(\tau \) we have \(u_{t}\in C^{\left( 1\right) }\left( X^{p}\right) ,\)
Moreover, by Lemma 3.3 we have \(u_{t}\in C^{\left( 1\right) }\left( X^{p}\right) \). The second statement follows similarly from (4.7).
Let
Condition 3.3
Assume that Condition 3.2 hold and A is a symmetric operator in H. Suppose that \(s+r\ge 1\) and \(a\left( x\right) =a\left( -x\right) \). Let \(\varphi ,\) \(\psi \in Y^{s,2}\left( A,H\right) \cap X_{\infty }\) and \(\Phi \left( .\right) \in L^{1}\)
Lemma 4.3
Let Condition 3.3 hold and let \(u\in C^{\left( 2,s\right) }\left( A,H\right) \) be a solution of (1.1)–(1.2) for any \(t\in \left[ 0,\right. \left. T\right) \). Then the energy
is constant.
Proof
By assumptions \(\Phi \left( .\right) \in L^{1}\) and Au, \( Au_{t}\) \(\in X^{2}\). Due to \(a\left( x\right) =a\left( -x\right) ,\) we have
Hence from (4.12) we obtain
where \(\left( u,\upsilon \right) \) denotes the inner product in \(X_{2}\). Hence, we obtain the assertion.
5 Blow-Up in Finite Time
In this section we prove the following result:
Theorem 5.1
Let Condition 3.3 hold and let \(u\in C^{\left( 2,s\right) }\left( A,H\right) \) be a solution of (1.1) –(1.2) for any \(t\in \left[ 0,\right. \left. T\right) \). If there exist positive numbers \(\nu \) and \(t_{0}\) such that
and
then the solution u blows up in finite time.
Proof
Assume that there is a global solution. Then u, \(u_{t}\in \) \(X_{2}\) for all \(t>0\). Let
for some b and \(t_{0}\) that will be determined later. We have
From (1.1), (5.1) , and (5.2) we get
Let b be a real number such that \(b\le -E\left( 0\right) \) and
From (5.3) and (5.5), we obtain
On the other hand, in view of the Cauchy-Schwarz inequality, we have
Hence, by (5.3), (5.5) , and (5.7) , we obtain
when
i.e., if assumption (5.2) holds. Moreover,
for sufficiently large \(t_{0}\). Then reasoning as in the proof of [16, Theorem1] , we get that that H(t), and thus \(\left\| u\left( t\right) \right\| ^{2}\) blows up in finite time, contradicting the assumption that a global solution exists.
6 Applications
6.1 The Cauchy Problem for Infinite System of WEs
Consider problem (1.4). Let
(see [27, § 1.18]). Let \(A_{1}\) be the operator in \( l_{2}\) defined by
Let
Let \(f=\left\{ f_{m}\right\} \), \(m=1,2,...,\infty \) and
Here
where
From Theorem 3.1 we obtain the following result:
Theorem 6.1
Assume the following: (1) assumption \(\left( 2.7\right) \) holds, \(0\le \alpha <1-\frac{1}{2p}\), \(\varphi \in \mathbb {H} _{0p}\left( l_{2}\right) \), \(\psi \in \) \(\mathbb {H}_{1p}\left( l_{2}\right) \) for \(p\in \left[ 1,\infty \right] \); (2) \(a\ge 0\), \(b_{j}\) are nonnegatıve bounded numbers, \(\hat{a}\left( \xi \right) +b_{j}>0\) for \(\xi \in \mathbb {R}^{n},\) and the following estimate hold
(3) the function
is measurable in \(\left( x,t\right) \in \mathbb {R}^{n}\times \left[ 0,T \right] \) for \(u\in W_{0}\left( l_{2}\right) \). Moreover, \(f\left( x,t,u\right) \) is continuous in \(u\in W_{0}\left( l_{2}\right) \) and \(f\in C^{\left[ s\right] +1}\left( W_{0}\left( l_{2}\right) ;l_{2}\right) \) uniformly in \(x\in \mathbb {R}^{n}\), \(t\in \left[ 0,T\right] \). Then problem \( \left( 1.3\right) \) has a unique local strong solution
where \(T_{0}\) is a maximal time interval that is appropriately small relative to M. Moreover, if
then \(T_{0}=\infty .\)
Proof
It is known that \(L^{p}\left( \mathbb {R}^{n};l_{2}\right) \) is a UMD space for \(p\in \left( 1,\infty \right) \) (se,e e.g., \(\left[ 25 \right] \)). By Remark 2.1, the definition of \(W^{s,p}\left( A_{1},l_{2}\right) \) and real interpolation of Banach spaces (see, e.g., [27, § 1.3, 1.18]), we have
By assumptions (1), (2) we obtain that \(A_{1}\) is sectorial in \(l_{2},\) and by virtue of [3, § 3.14, 3.16], the operator \(A_{1}^2+\mu \) is a generator of bounded cosine function in \(l_{2}\). Hence, by (4), (5), all conditions of Theorem 3.2 are satisfied, i.e., we get the conclusion.
Theorem 6.2
Assume: (h\(_{1}\)) assumptions (1)-(3) of Theorem 6.1 are satisfied for \(p=2\); (h\(_{2}\)) \(f_{m}\in C^{\left[ s\right] }\left( \mathbb {R};l_{2}\right) \) with \(f(0)=0\) and
(h\(_{3}\)) \(B\varphi \), \(B\psi \in L^{2}\left( \mathbb {R}^{n};l_{2}\right) \) and \(\Phi \left( \varphi \right) \in L^{2}\left( \mathbb {R}^{n};l_{2}\right) \); (h\(_{4}\)) there is some \(k>0\) so that
Then: (a); there exists \(T>0\) such that problem (1.4) has a global solution
(b) if assumption 5.1 of Theorem 5.1 also holds for \(H=l_{2}\), then the solution of (1.4) blows up in finite time.
Proof
From assumptions (h\(_{1}\)), (h\(_{2}\)) it is clear to that Condition 4.1 holds for \(H=l_{2}\) and \(r>2+\frac{n}{2}\). By (h\(_{3}\)), all other assumptions of Theorem 4.1 are satisfied. Hence, we obtain the assertion.
6.2 The Mixed Problem for Degenerate WE
Consider the problem (1.5)–(1.7). Let
Let \(A_{2}\) be the operator in \(L^{2}\left( 0,1\right) \) defined by (1.6)–(1.8) and let
Here
where
Now we present the following result:
Condition 6.1
Assume;
(1) assumption (2.7) holds, \(0\le \gamma <\frac{1}{2},\) and \( \alpha _{1}\beta _{2}-\alpha _{2}\beta _{1}\ne 0;\)
(2) \(0\le \alpha <1-\frac{1}{2p}\), \(\varphi \in \mathbb {H}_{0p}\left( L^{2}\right) \), \(\psi \in \) \(\mathbb {H}_{1p}\left( L^{2}\right) \) for \(p\in \left[ 1,\infty \right] \);
(2) \(b_{1}\) and \(b_{2}\) are complex valued functions on \(\left( 0,1\right) \) . Moreover, \(b_{1}\in C\left[ 0,1\right] ,\) \(b_{1}\left( 0\right) =b_{1}\left( 1\right) \), \(b_{2}\in L_{\infty }\left( 0,1\right) ,\) and \( \left| b_{2}\left( x\right) \right| \le C\) \(\left| b_{1}^{\frac{ 1}{2}-\mu }\left( x\right) \right| \) for \(0<\mu <\frac{1}{2}\) and for a.a. \(x\in \left( 0,1\right) ;\)
(3) \(a\ge 0\) and \(\eta _{2}\left( \xi \right) \ne 0\) for all \(\xi \in \mathbb {R}^{n};\)
(4) the function
is measurable in \(\left( x,t\right) \in \mathbb {R}^{n}\times \left[ 0,T \right] \) for \(u\in W_{0}\left( L^{2}\left( 0,1\right) \right) \); \(f\left( x,t,u\right) \). Moreover, \(f\left( x,t,u\right) \) is continuous in \(u\in W_{0}\left( L^{2}\left( 0,1\right) \right) \) and
uniformly with respect to \(x\in \mathbb {R}^{n}\), \(t\in \left[ 0,T\right] .\)
Theorem 6.3
Assume that Condition 6.1 is satisfied. Then problem (1.6)–(1.8) has a unique local strong solution
where \(T_{0}\) is a maximal time interval that is appropriately small relative to M. Moreover, if
then \(T_{0}=\infty .\)
Proof
It is known (see, e.g., [13]) that \( L^{2}\left( 0,1\right) ,\) is a UMD space for \(p_{1}\in \left( 1,\infty \right) \). By definition, \(W^{s,p}\left( A_{2},L^{2}\left( 0,1\right) \right) ,\) and by real interpolation of Banach spaces (see, e.g., [27, §1.3.2.]) we have
In view of [26, Theorem 4.1], we obtain that the operator \(A_{2}\) defined by (2.5) is uniformly sectorial in \( L^{2}\left( 0,1\right) ,\) and by virtue of [3, § 3.14, 3.16], the operator \(A_{2}^2+\mu \) is a generator of the bounded cosine function in \(L^{2}\left( 0,1\right) \). Moreover, using assumptions (1), (2), we deduce that \(\eta _{2}\left( \xi \right) \ne 0\) for all \(\ \xi \in \mathbb {R}^{n}\). Hence by hypotheses (3), (4) of Condition 5.1, we get that all, hypotheses of Theorem 3.2 hold, i.e., we obtain the conclusion.
Theorem 6.4
Assume Condition 6.1 is satisfied for \( p_{1}=2 \). Suppose \(f\in C^{[ s] }(\mathbb {R};L^{2}((0,1))) \) with \(f(0)=0\). Moreover, let \(\ B\varphi \), \(B\psi \), \(\Phi (\varphi ) \in L^{2}(\mathbb {R} ^{n}\times (0,1)) \) and there exists \(k>0\) such that
Then:
(a) there exists \(T>0\) such that problem \((1.5)-\left( 1.7\right) \) has a global solution
(b) if assumption (5.1) of the Theorem 5.1 also holds for \(H=L^{2}\left( 0,1\right) \), then the solution of \(\left( 1.6\right) -\left( 1.8\right) \) blows up in finite time.
Proof
Indeed, by assumption, all conditions of Theorem 4.1 are satisfied for \(H=L^{2}\left( 0,1\right) \), i.e., we obtain the assertion.
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Shakhmurov, V., Shahmurov, R. The Regularity Properties and Blow-up of Solutions for Nonlocal Wave Equations and Applications. Results Math 77, 229 (2022). https://doi.org/10.1007/s00025-022-01752-y
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DOI: https://doi.org/10.1007/s00025-022-01752-y
Keywords
- Abstract differential equations
- regularity properties
- wave equations
- blow-up of solutions
- fourier multipliers