<p>We study a class of impulsive pseudo-parabolic equations with the nonlinear source terms depending on the solution, its gradient, and Laplacian. The smooth coefficient <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varphi _n(t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>φ</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> before the source term has the support in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$[0,\frac{1}{n}]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> and converges to the Dirac delta function <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\delta _{(t=0)}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>δ</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msub> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$n\rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. It is shown that the sequence of solutions of the non-instantaneous impulsive equations converges as <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$n\rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> to a solution of the instantaneous impulsive equation, and that the new initial datum is generated by the solution of a third-order equation on the infinitesimal initial layer.</p>

Impulsive pseudo-parabolic equations with nonstandard growth, nonlinear source term and infinitesimal initial layer

期刊论文
发表日期：

2026年1月8日

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Stanislav Antontsev,
Ivan Kuznetsov,
Sergey Shmarev

摘要

We study a class of impulsive pseudo-parabolic equations with the nonlinear source terms depending on the solution, its gradient, and Laplacian. The smooth coefficient $\varphi _n(t)$ $φ_{n} (t)$ before the source term has the support in $[0,\frac{1}{n}]$ $[0, \frac{1}{n}]$ and converges to the Dirac delta function $\delta _{(t=0)}$ $δ_{(t = 0)}$ as $n\rightarrow \infty $ $n \to \infty$ . It is shown that the sequence of solutions of the non-instantaneous impulsive equations converges as $n\rightarrow \infty $ $n \to \infty$ to a solution of the instantaneous impulsive equation, and that the new initial datum is generated by the solution of a third-order equation on the infinitesimal initial layer.