On the Solvability of the Boundary-Value Problem for the First Order Operator-Differential Equations with Multiple Characteristics

Mohamed A. Labeeb¹, Ahmed L. Elbably², Abdel Baset I. Ahmed¹

¹Egyptian Russian University, Badr City, Egypt.
²Helwan University, Helwan, Egypt.

DOI: 10.4236/oalib.1104155                

Abstract

In this paper, a class of operator-differential equation of the first order with multiple characteristics is considered, for which the initial boundary value problem on the semi-axis is well-posed and uniquely solvable in the Sobolev space.

Keywords

Operator-Differential Equation, Boundary-Value Problem, Hilbert Space, Reg-ular Solvability

Suppose that H is a separable Hilbert space with a scalar product $\left(x\mathrm{,}y\right)$ , $x\mathrm{,}y\in H$ , A is a self-adjoint non negative definite operator on H ( $A=A^{*}\ge cE,c>0$ , E is the identity operator). By $H_{\gamma }$ ( $\gamma \ge 0$ ), we denote the scale of Hilbert spaces generated by the operator A, i.e. $H_{\gamma }=Dom\left(A^{\gamma }\right)$ , ${\left(x\mathrm{,}y\right)}_{\gamma }=\left(A^{\gamma }x\mathrm{,}{A}^{\gamma }y\right)$ , $x\mathrm{,}y\in Dom\left(A^{\gamma }\right)$ . For $\gamma \mathrm{<mo>\; =\; </mo>0}$ we consider that $H_0=H$ , ${\left(x\mathrm{,}y\right)}_0=\left(x\mathrm{,}y\right)$ , $x\mathrm{,}y\in H$ .

By $L_2\left(\left[e\mathrm{,}b\right]\mathrm{<mo>\; ;\; </mo>}H\right)$ ( $-\infty \le e\mathrm{<mo>\; <\; </mo>}b\le +\infty$ ), we denote the space of measurable functions (see [1] ) with values in H and the norm

${\Vert f\Vert }_{L_2\left(\left[a\mathrm{,}b\right]\mathrm{<mo>\; ;\; </mo>}H\right)}={\left({\displaystyle {\int }_e^b}{\Vert f\left(t\right)\Vert }^2\text{d}t\right)}^{1/2}\mathrm{,}$

$W_2^1\left(\left[e\mathrm{,}b\right]\mathrm{<mo>\; ;\; </mo>}H\right)$ is the space of all functions with values in H such that $\frac{\text{d}u\left(t\right)}{\text{d}t}$ ,

$Au\left(t\right)\in L_2\left(\left[e\mathrm{,}b\right]\mathrm{<mo>\; ;\; </mo>}H\right)$ with the norm

${\Vert u\Vert }_{W_2^1\left(\left[e\mathrm{,}b\right]\mathrm{<mo>\; ;\; </mo>}H\right)}={\left({\Vert \frac{\text{d}u}{\text{d}t}\Vert }_{L_2\left(\left[a\mathrm{,}b\right]\mathrm{<mo>\; ;\; </mo>}H\right)}^2+{\Vert Au\Vert }_{L_2\left(\left[a\mathrm{,}b\right]\mathrm{<mo>\; ;\; </mo>}H\right)}^2\right)}^{1/2}\mathrm{.}$

For more details about the space $W_2^1\left(\left[e\mathrm{,}b\right]\mathrm{<mo>\; ;\; </mo>}H\right)$ , see [ [2] , Ch.1]. We assume that

For $e=-\infty$ , $b=+\infty$

$L_2\left(\left(-\infty ,+\infty \right);H\right)\equiv L_2\left(R;H\right),W_2^1\left(\left(-\infty ,+\infty \right);H\right)\equiv W_2^1\left(R;H\right),R=\left(-\infty ,+\infty \right),$

For $e=0$ , $b=+\infty$

$L_2\left(\left[0,+\infty \right);H\right)\equiv L_2\left(R_{+};H\right),W_2^1\left(\left[0,+\infty \right);H\right)\equiv W_2^1\left(R_{+};H\right),R_{+}=\left[0,+\infty \right).$

For the whole article, all the derivatives are understood in the sense of distributions.

Consider the following problem on the semiaxis $R_{+}$ :

$\frac{\text{d}u\left(t\right)}{\text{d}t}+Au\left(t\right)=f\left(t\right),t\in R_{+},$ (1)

$u\left(0\right)=0,$ (2)

where $A=A^{*}\ge cE,c>0$ , $f\left(t\right)\in L_2\left(R_{+}\mathrm{<mo>\; ;\; </mo>}H\right)$ , $u\left(t\right)\in W_2^1\left(R_{+}\mathrm{<mo>\; ;\; </mo>}H\right)$ .

Definition 1. If the vector function $u\left(t\right)\in W_2^1\left(R_{+}\mathrm{<mo>\; ;\; </mo>}H\right)$ satisfies Equation (1) nearly everywhere in $R_{+}$ , then it is called a regular solution of Equation (1).

Definition 2. For any $f\left(t\right)\in L_2\left(R_{+}\mathrm{<mo>\; ;\; </mo>}H\right)$ there exists a regular solution $u\left(t\right)$ of Equation (1) satisfying the boundary Conditions (2) in the sense of relation

$\underset{t\to 0}{lim}\Vert A^{1/2}u\left(t\right)\Vert =\mathrm{0,}$

and the following inequality holds:

${\Vert u\Vert }_{W_2^1\left(\left[e\mathrm{,}b\right]\mathrm{<mo>\; ;\; </mo>}H\right)}\le \text{const}{\Vert f\Vert }_{L_2\left(R_{+}\mathrm{<mo>\; ;\; </mo>}H\right)}\mathrm{,}$

Then we say that say that problem (1), (2) is regularly solvable.

In this paper, the regular solvability of the initial boundary value problem (1), (2) is established.

In spite of the fact that, since 50th years of the last century, there are sufficiently many papers and books dedicated to the solvability of parabolic operator-differential equations in Banach spaces, in particular, Hilbert spaces, interest in such equations doesn’t abate until recently (see, for example, [3] [4] [5] ). However, in the mathematical literature studies on the parabolic operator-differential equations with multiple characteristics are almost not mentioned, although they have a wide application in some problems in mechanics and mathematical physics. Note that Equation (1) is an abstract parabolic equation in Hilbert space that has multiple characteristics.

We let

$\stackrel{o}{{\displaystyle W_2^1}}\left(R_{+};H\right)=\left\{u\left(t\right):u\left(t\right)\in W_2^1\left(R_{+};H\right),u\left(0\right)=0\right\}.$

And denote by $Q_0$ the operator acting from the space $\stackrel{o}{{\displaystyle W_2^1}}\left(R_{+}\mathrm{<mo>\; ;\; </mo>}H\right)$ to the

space $L_2\left(R_{+}\mathrm{<mo>\; ;\; </mo>}H\right)$ as follows:

$Q_{0}u\left(t\right)\equiv \frac{\text{d}u\left(t\right)}{\text{d}t}+Au\left(t\right),u\left(t\right)\in \stackrel{o}{{\displaystyle W_2^1}}\left(R_{+};H\right).$

Then the following theorem holds

Theorem 1. The operator $Q_0$ is an isomorphism between the spaces $\stackrel{o}{{\displaystyle W_2^1}}\left(R_{+}\mathrm{<mo>\; ;\; </mo>}H\right)$ and $L_2\left(R_{+}\mathrm{<mo>\; ;\; </mo>}H\right)$ .

Proof. Obviously, the equation $Q_{0}u\left(t\right)=0$ , has only the trivial solution

$u\left(t\right)\in \stackrel{o}{{\displaystyle W_2^1}}\left(R_{+}\mathrm{<mo>\; ;\; </mo>}H\right)$ . Really, the equation $Q_{0}u\left(t\right)=0$ has a solution in the space

$W_2^1\left(R_{+}\mathrm{<mo>\; ;\; </mo>}H\right)$ of the form $u_0\left(t\right)={\text{e}}^{-tA}\phi$ , where the vectors $\phi \in H_{1/2}$ . Taking into account conditions (??), we have $\phi =0$ and $u_0\left(t\right)=0$ . Therefore, the equation

$Q_{0}u\left(t\right)=0$ has only the trivial solution in the space $\stackrel{o}{{\displaystyle W_2^1}}\left(R_{+}\mathrm{<mo>\; ;\; </mo>}H\right)$ .

Now we shall show that the equation $Q_{0}u\left(t\right)=f\left(t\right)$ for any $f\left(t\right)\in L_2\left(R_{+}\mathrm{<mo>\; ;\; </mo>}H\right)$

has a solution in the space $\stackrel{o}{{\displaystyle W_2^1}}\left(R_{+}\mathrm{<mo>\; ;\; </mo>}H\right)$ . We extend the function $f\left(t\right)$ to

zero for $t<0$ , then, applying the direct and inverse Fourier transform, it becomes clear that

$u_1\left(t\right)=\frac{1}{2\text{π}}{\displaystyle {\int }_{-\infty }^{+\infty }}{\left(i\xi E+A\right)}^{-1}\left({\displaystyle {\int }_0^{+\infty }}f\left(s\right){\text{e}}^{-i\xi s}\text{d}s\right){\text{e}}^{i\xi t}\text{d}\xi ,t\in R,$

Satisfies the equation $Q_{0}u\left(t\right)=f\left(t\right)$ almost everywhere in R. We show that $u_1\left(t\right)\in W_2^1\left(R\mathrm{<mo>\; ;\; </mo>}H\right)$ . From Parseval’s equality we obtain:

$\begin{array}{c}{\Vert u_1\Vert }_{W_2^2\left(R;H\right)}^2={\Vert \frac{\text{d}{u}_1}{\text{d}t}\Vert }_{L_2\left(R;H\right)}^2+{\Vert A{u}_1\Vert }_{L_2\left(R;H\right)}^2\\ ={\Vert i\xi {\tilde{u}}_1\left(\xi \right)\Vert }_{L_2\left(R;H\right)}^2+{\Vert A{\tilde{u}}_1\left(\xi \right)\Vert }_{L_2\left(R;H\right)}^2\\ ={\Vert i\xi {\left(i\xi E+A\right)}^{-1}\tilde{f}\left(\xi \right)\Vert }_{L_2\left(R;H\right)}^2+{\Vert A{\left(i\xi E+A\right)}^{-1}\tilde{f}\left(\xi \right)\Vert }_{L_2\left(R;H\right)}^2\\ \le \underset{\xi \in R}{\sup }{\Vert i\xi {\left(i\xi E+A\right)}^{-1}\Vert }_{H\to H}^2{\Vert \tilde{f}\left(\xi \right)\Vert }_{L_2\left(R;H\right)}^2\\ +\underset{\xi \in R}{\sup }{\Vert A{\left(i\xi E+A\right)}^{-1}\Vert }_{H\to H}^2{\Vert \tilde{f}\left(\xi \right)\Vert }_{L_2\left(R;H\right)}^2,\end{array}$ (3)

where ${\tilde{u}}_1\left(\xi \right)$ and $\tilde{f}\left(\xi \right)$ are the Fourier transforms of the functions $u_1\left(t\right)$ and $f\left(t\right)$ , respectively. $\sigma \left(A\right)$ denotes the spectrum of the operator A. From the spectral decomposition of the operator A for $\xi \in R$ , we have:

$\Vert i\xi {\left(i\xi E+A\right)}^{-1}\Vert =\underset{\mu \in \sigma \left(A\right)}{\sup }\left|i\xi {\left(i\xi +\mu \right)}^{-1}\right|\le \underset{\mu \in \sigma \left(A\right)}{\sup }\frac{\left|\xi \right|}{{\left({\xi }^2+{\mu }^2\right)}^{\frac{1}{2}}}\le 1,$ (4)

$\Vert A_{}{\left(i\xi E+A\right)}^{-1}\Vert =\underset{\mu \in \sigma \left(A\right)}{\sup }\left|\mu {\left(i\xi +\mu \right)}^{-1}\right|\le \underset{\mu \in \sigma \left(A\right)}{\sup }{\left(\frac{{\mu }^2}{{\xi }^2+{\mu }^2}\right)}^{\frac{1}{2}}\le 1.$ (5)

Taking into account (4) and (5) into (3) we obtain:

${\Vert u_1\Vert }_{W_2^1\left(\left[e,b\right];H\right)}^2\le 2{\Vert \tilde{f}\left(\xi \right)\Vert }_{L_2\left(R;H\right)}^2=2{\Vert f\left(t\right)\Vert }_{L_2\left(R_{+};H\right)}^2.$

Therefore, $u_1\left(t\right)\in W_2^1\left(R\mathrm{<mo>\; ;\; </mo>}H\right)$ . Now we denote the restriction of the function $u_1\left(t\right)$ to $R_{+}$ by $v\left(t\right)$ . Obviously, $v\left(t\right)\in W_2^1\left(R_{+}\mathrm{<mo>\; ;\; </mo>}H\right)$ . By the theorem on traces [ [2] , Ch.1] $v\left(0\right)\in H_{1/2}$ . Then we will find a solution to the equation $Q_{0}u\left(t\right)=f\left(t\right)$ of the form

$u\left(t\right)=v\left(t\right)+{\text{e}}^{-tA}\psi ,$

where the vectors $\psi \in H_{1/2}$ . For a clear determination of $\psi$ from conditions (2) we obtain that $v\left(0\right)+\psi =0$ ,

We have

$\psi =-v\left(0\right)\in H_{1/2},$

Thus,

$u\left(t\right)\in \stackrel{o}{{\displaystyle W_2^1}}\left(R_{+}\mathrm{<mo>\; ;\; </mo>}H\right)\mathrm{.}$

It is clear that the operator $Q_0$ is bounded and acts from the space $\stackrel{o}{{\displaystyle W_2^1}}\left(R_{+}\mathrm{<mo>\; ;\; </mo>}H\right)$ to the space $L_2\left(R_{+}\mathrm{<mo>\; ;\; </mo>}H\right)$ . Indeed, taking into account the theorem on intermediate derivatives [ [2] , Ch.1], we have:

$\begin{array}{c}{\Vert Q_{0}u\Vert }_{L_2\left(R_{+};H\right)}={\Vert \frac{\text{d}u}{\text{d}t}+Au\Vert }_{L_2\left(R_{+};H\right)}\le {\Vert \frac{\text{d}u}{\text{d}t}\Vert }_{L_2\left(R_{+};H\right)}+{\Vert Au\Vert }_{L_2\left(R_{+};H\right)}\\ \le \text{const}{\Vert u\Vert }_{W_2^1\left(R_{+};H\right)}.\end{array}$

Thus, the operator $Q_0\mathrm{<mo>\; :\; </mo>}\stackrel{o}{{\displaystyle W_2^1}}\left(R_{+}\mathrm{<mo>\; ;\; </mo>}H\right)\to L_2\left(R_{+}\mathrm{<mo>\; ;\; </mo>}H\right)$ is bijective and bounded. Hence, using the Banach inverse operator theorem, the operator $Q_0^{-1}\mathrm{<mo>\; :\; </mo>}{L}_2\left(R_{+}\mathrm{<mo>\; ;\; </mo>}H\right)\to \stackrel{o}{{\displaystyle W_2^1}}\left(R_{+}\mathrm{<mo>\; ;\; </mo>}H\right)$ is bounded. Therefore, the operator $Q_0$ is an isomorphism between the spaces $\stackrel{o}{{\displaystyle W_2^1}}\left(R_{+}\mathrm{<mo>\; ;\; </mo>}H\right)$ and $L_2\left(R_{+}\mathrm{<mo>\; ;\; </mo>}H\right)$ .

The theorem is proved. □

Corollary 2. It follows from Theorem 1 that initial-boundary value problem (1), (2) is regularly solvable.

Corollary 3. It follows from Theorem 1 that the norms ${\Vert u\Vert }_{W_2^1\left(R_{+}\mathrm{<mo>\; ;\; </mo>}H\right)}$ and ${\Vert Q_{0}u\Vert }_{L_2\left(R_{+}\mathrm{<mo>\; ;\; </mo>}H\right)}$ are equivalent on the space $\stackrel{o}{{\displaystyle W_2^1}}\left(R_{+}\mathrm{<mo>\; ;\; </mo>}H\right)$ .

Conflicts of Interest

The authors declare no conflicts of interest.

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