In this paper we study the third-order abstract ordinary differential equations <Equation ID="Equ21"> <EquationSource Format="TEX">$\begin{aligned} u_{ttt}+aA^{x}u_{tt}+bA^{y}u_t+c^2A^zu=0, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>u</mi> <mrow> <mi mathvariant="italic">ttt</mi> </mrow> </msub> <mo>+</mo> <mi>a</mi> <msup> <mi>A</mi> <mi>x</mi> </msup> <msub> <mi>u</mi> <mrow> <mi mathvariant="italic">tt</mi> </mrow> </msub> <mo>+</mo> <mi>b</mi> <msup> <mi>A</mi> <mi>y</mi> </msup> <msub> <mi>u</mi> <mi>t</mi> </msub> <mo>+</mo> <msup> <mi>c</mi> <mn>2</mn> </msup> <msup> <mi>A</mi> <mi>z</mi> </msup> <mi>u</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$a,b\geqslant 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>⩾</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$c\ne 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo>≠</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$0\leqslant x<y\leqslant z\leqslant 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>⩽</mo> <mi>x</mi> <mo><</mo> <mi>y</mi> <mo>⩽</mo> <mi>z</mi> <mo>⩽</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$A:D(A)\subset X \rightarrow X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>:</mo> <mi>D</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>⊂</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation> is an unbounded, closed, densely defined, self-adjoint linear operator, which is defined on a separable Hilbert space X. We characterized the spectrum of the linear operator associated to the equation and analyzed for which coefficients a, b, c and for which powers x, y, z, this system can generate a strongly continuous semigroup, or an analytic semigroup, or does not generate anything. Since the coefficients have an important role when it comes to differential equations of third-order, we appeal to a tool from Galois theory, namely the discriminant of a polynomial. Finally, we compute the probabilities, in some sense, of this equation having a solution (strongly continuous semigroup) and being regular (analytic semigroup).

Spectral and Probabilistic Analysis of Third-Order Linear Abstract Differential Equations

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发表日期：

2025年3月17日

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Flank D. M. Bezerra,
Heraclio López-Lázaro,
Carlos R. Takaessu Jr.

摘要

In this paper we study the third-order abstract ordinary differential equations $\begin{aligned} u_{ttt}+aA^{x}u_{tt}+bA^{y}u_t+c^2A^zu=0, \end{aligned}$ $\begin{matrix} u_{ttt} + a A^{x} u_{tt} + b A^{y} u_{t} + c^{2} A^{z} u = 0, \end{matrix}$ where $a,b\geqslant 0$ $a, b ⩾ 0$ , $c\ne 0$ $c \neq 0$ , $0\leqslant x<y\leqslant z\leqslant 1$ $0 ⩽ x < y ⩽ z ⩽ 1$ , and $A:D(A)\subset X \rightarrow X$ $A : D (A) \subset X \to X$ is an unbounded, closed, densely defined, self-adjoint linear operator, which is defined on a separable Hilbert space X. We characterized the spectrum of the linear operator associated to the equation and analyzed for which coefficients a, b, c and for which powers x, y, z, this system can generate a strongly continuous semigroup, or an analytic semigroup, or does not generate anything. Since the coefficients have an important role when it comes to differential equations of third-order, we appeal to a tool from Galois theory, namely the discriminant of a polynomial. Finally, we compute the probabilities, in some sense, of this equation having a solution (strongly continuous semigroup) and being regular (analytic semigroup).