We investigate existence of global in time solutions versus blow-up ones for the semilinear heat equation posed on infinite graphs. The source term is a general function f(u), and the different behaviour of solutions is characterized by the behaviour of f near the origin and by the first eigenvalue <InlineEquation ID="IEq1"> <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> of the negative Laplacian on the graph, which is assumed to satisfy <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\lambda _1(G)>0$</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> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. In particular, if <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$f'(0)>\lambda _1(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>></mo> <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> than all positive nontrivial solution blows up in finite time, whereas if <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$f'(0)<\lambda _1(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo><</mo> <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>, or if a weaker condition involving the Lipschitz constant of f in a neighborhood of the origin holds, then there exist global in time, bounded solutions.

Blow-up and global existence for semilinear parabolic equations on infinite graphs

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2026年2月28日

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Gabriele Grillo,
Giulia Meglioli,
Fabio Punzo

摘要

We investigate existence of global in time solutions versus blow-up ones for the semilinear heat equation posed on infinite graphs. The source term is a general function f(u), and the different behaviour of solutions is characterized by the behaviour of f near the origin and by the first eigenvalue $\lambda _1(G)$ $λ_{1} (G)$ of the negative Laplacian on the graph, which is assumed to satisfy $\lambda _1(G)>0$ $λ_{1} (G) > 0$ . In particular, if $f'(0)>\lambda _1(G)$ $f^{'} (0) > λ_{1} (G)$ than all positive nontrivial solution blows up in finite time, whereas if $f'(0)<\lambda _1(G)$ $f^{'} (0) < λ_{1} (G)$ , or if a weaker condition involving the Lipschitz constant of f in a neighborhood of the origin holds, then there exist global in time, bounded solutions.