We are concerned with semilinear parabolic equations, with a time-dependent source term of the form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$h(t)u^q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>h</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>u</mi> <mi>q</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$q>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, posed on an infinite graph. We assume that the bottom of the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-spectrum of the Laplacian on the graph, denoted by <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\lambda _1(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, is positive. In dependence of q, h(t) and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\lambda _1(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, we show global in time existence or finite time blow-up of solutions.

On a semilinear parabolic equation with time-dependent source term on infinite graphs

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2026年1月18日

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Fabio Punzo,
Alessandro Sacco

摘要

We are concerned with semilinear parabolic equations, with a time-dependent source term of the form $h(t)u^q$ $h (t) u^{q}$ with $q>1$ $q > 1$ , posed on an infinite graph. We assume that the bottom of the $L^2$ $L^{2}$ -spectrum of the Laplacian on the graph, denoted by $\lambda _1(G)$ $λ_{1} (G)$ , is positive. In dependence of q, h(t) and $\lambda _1(G)$ $λ_{1} (G)$ , we show global in time existence or finite time blow-up of solutions.