This paper is concerned with the life span of solutions for a semilinear pseudo-parabolic equation with inhomogeneous source term <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\lambda f(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. First, by using Kaplan’s first eigenvalue method, an upper bound estimate for the life span <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ T_\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>λ</mi> </msub> </math></EquationSource> </InlineEquation> is obtained. Based on this, by means of the properties of the pseudo-parabolic kernel, we establish the asymptotic behavior of the life span <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ T_\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>λ</mi> </msub> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ \lambda \rightarrow 0 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> for the case where f(x) is a radially decreasing function. Finally, employing the comparison principle, we obtain the asymptotic behavior of the life span <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ T_\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>λ</mi> </msub> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$ \lambda \rightarrow 0 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>.

Life span of solutions for a semilinear pseudo-parabolic equation with inhomogeneous source

期刊论文
发表日期：

2026年6月3日

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Yang Cao,
Benhui Wang

摘要

This paper is concerned with the life span of solutions for a semilinear pseudo-parabolic equation with inhomogeneous source term $\lambda f(x)$ $λ f (x)$ . First, by using Kaplan’s first eigenvalue method, an upper bound estimate for the life span $ T_\lambda $ $T_{λ}$ is obtained. Based on this, by means of the properties of the pseudo-parabolic kernel, we establish the asymptotic behavior of the life span $ T_\lambda $ $T_{λ}$ as $ \lambda \rightarrow 0 $ $λ \to 0$ for the case where f(x) is a radially decreasing function. Finally, employing the comparison principle, we obtain the asymptotic behavior of the life span $ T_\lambda $ $T_{λ}$ as $ \lambda \rightarrow 0 $ $λ \to 0$ .