Well-Posedness and Spectral Analysis of Integrodifferential Equations of Hereditary Mechanics

Abstract

The well-posedness of initial value problems for abstract integrodifferential equations with unbounded operator coefficients in Hilbert spaces is studied, and spectral analysis of the operator functions that are the symbols of these equations is performed. The equations under consideration are an abstract form of linear partial integrodifferential equations arising in viscoelasticity theory, which have a number of other important applications. Results concerning the well-posedness of these integrodifferential equations in weighted Sobolev spaces of vector functions defined on the positive half-line with values in a Hilbert space are obtained. The localization and structure of the spectrum of the operator functions that are the symbols of these equations are established.

APPENDIX

Proof of Lemma 3.3. Let $f \in {{L}_{{2,\gamma }}}({{\mathbb{R}}_{ + }},H)$. We seek a function $h\left( t \right)$ such that $h(t) \in W_{{2,\gamma }}^{1}({{\mathbb{R}}_{ + }},A_{0}^{{1/2}}),$ $h(0) = 0$, and $\mathop {\left\| {f(t) - h(t)} \right\|}\nolimits_{{{L}_{{2,\gamma }}}({{\mathbb{R}}_{ + }},H)} \leqslant \varepsilon .$ Let $g(t): = {{e}^{{ - \gamma t}}}f(t)$, $g(t) \in {{L}_{2}}({{\mathbb{R}}_{ + }},H)$, and let $\theta (t): = {{e}^{{ - \gamma t}}}h(t)$, $\theta (t) \in W_{2}^{1}({{\mathbb{R}}_{ + }},A_{0}^{{1/2}}t)$. The following chain of equalities holds:

$$\mathop {\left\| {f(t) - h(t)} \right\|}\nolimits_{{{L}_{{2,\gamma }}}({{R}_{ + }},H)}^2 = \int\limits_0^\infty {\mathop {{{e}^{{ - 2\gamma t}}}\left\| {f(t) - h(t)} \right\|}\nolimits^2 dt} = \int\limits_0^\infty {{{{\left\| {{{e}^{{ - \gamma t}}}f(t) - {{e}^{{ - \gamma t}}}h(t)} \right\|}}^{2}}dt} = \int\limits_0^\infty {{{{\left\| {g(t) - \theta (t)} \right\|}}^{2}}dt} .$$

Thus, the problem of the denseness, in the space ${{L}_{{2,\gamma }}}({{\mathbb{R}}_{ + }},H)$, of the family of functions such that $h(t) \in W_{{2,\gamma }}^{1}({{\mathbb{R}}_{ + }},A_{0}^{{1/2}})$ and $h\left( { + 0} \right) = 0$ reduces to the problem of the denseness in ${{L}_{2}}({{\mathbb{R}}_{ + }},H)$ of the family of functions $\{ \theta (t)\}$ such that $\theta (t) \in W_{2}^{1}({{\mathbb{R}}_{ + }},A_{0}^{{1/2}})$ and $\theta \left( { + 0} \right) = 0$. In turn, the denseness of $\{ \theta (t)\}$ is implied by the well-known result in [15]. In fact, according to Theorem 2.1 in [15], the family of infinitely differentiable compactly supported vector functions is everywhere dense in ${{L}_{2}}({{\mathbb{R}}_{ + }},H).$ Therefore, the subset of infinitely differentiable functions with support in ${{\mathbb{R}}_{ + }}$ is everywhere dense in ${{L}_{2}}({{\mathbb{R}}_{ + }},H)$. This family evidently belongs to the family of functions $\{ \theta (t)\}$ such that $\theta (t) \in W_{2}^{1}({{\mathbb{R}}_{ + }},A_{0}^{{1/2}})$ and $\theta \left( { + 0} \right) = 0$, which proves Lemma 3.3.

Proof of Lemma 3.4. We establish inequalities (3.8) and (3.9). The operator function ${{L}^{{ - 1}}}(\lambda )$ is representable as

$${{L}^{{ - 1}}}(\lambda ) = {{({{\lambda }^{2}}I + {{A}_{0}})}^{{ - 1}}}{{\left( {I - V(\lambda )} \right)}^{{ - 1}}},$$

(A1)

where the operator function $V(\lambda )$ is defined by (3.5). We can directly verify that

$$\left\| {A_{0}^{{1/2}}{{{({{\lambda }^{2}}I + {{A}_{0}})}}^{{ - 1}}}} \right\| \leqslant \frac{{{\text{const}}}}{{\left| {\operatorname{Re} \lambda } \right|}},\quad \left\| {\lambda {{{({{\lambda }^{2}}I + {{A}_{0}})}}^{{ - 1}}}} \right\| \leqslant \frac{{{\text{const}}}}{{\left| {\operatorname{Re} \lambda } \right|}}.$$

(A2)

Note that the second inequality in (A2) has been proved in Lemma 3.2. To prove the first inequality, according to the spectral theorem (see [14 , pp. 452–453]), it suffices to show that

$$\mathop {\sup}\limits_{\operatorname{Re} \lambda > \gamma } \left( {\frac{{\sqrt \alpha }}{{{{\lambda }^{2}} + \alpha }}} \right) \leqslant \frac{{{\text{const}}}}{{\left| {\operatorname{Re} \lambda } \right|}},\quad \alpha \in \sigma ({{A}_{0}}) \subset [{{\kappa }_{0}}, + \infty ).$$

(3.29)

Considering the real and imaginary parts of $\lambda = \tau + i\nu$, we arrive at the required inequality

$$\frac{{\sqrt \alpha }}{{\sqrt {{{\tau }^{2}} + {{{(\nu - \sqrt \alpha )}}^{2}}} \sqrt {{{\tau }^{2}} + {{{(\nu + \sqrt \alpha )}}^{2}}} }} \leqslant \frac{{{\text{const}}}}{{\left| \tau \right|}}.$$

Based on (A1), (A2), and Lemma 3.1, we derive the assertion of Lemma 3.4.