<p>We continue to study a free boundary problem <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$u_t=du_{xx}+f(u)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mi>t</mi> </msub> <mo>=</mo> <mi>d</mi> <msub> <mi>u</mi> <mrow> <mi mathvariant="italic">xx</mi> </mrow> </msub> <mo>+</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$t>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$0<x<h(t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>x</mi> <mo><</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with positive bistable nonlinearity <i>f</i> under Dirichlet boundary conditions, for which Endo et al. (Discrete Contin Dyn Syst 40:3375–3394, 2020) classified large-time behaviors of solutions into three groups: the big spreading, small spreading and vanishing. Our first aim is to classify the small spreading solutions into two types; one is a standard small spreading solution and the other is a spreading solution with a moving one-peak. The latter type of spreading solution will be understood to play a borderline-behavior between the big and small spreading. When we discuss large-time behaviors of spreading solutions, it is known that a solution of the corresponding semi-wave problem allows us to obtain sharp asymptotic profiles and estimates for spreading solutions. However, the semi-wave problem for a big spreading solution has no solutions under certain circumstances. Our second aim is to derive sharp asymptotic profiles and estimates for any big spreading solution even in this situation. We will show that such a big spreading solution converges as <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$t\rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> to a so called propagating terrace which consists of a stationary solution for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$0\le x\le c_1t$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>x</mi> <mo>≤</mo> <msub> <mi>c</mi> <mn>1</mn> </msub> <mi>t</mi> </mrow> </math></EquationSource> </InlineEquation>, a travelling wave for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$c_1t\le x\le c_2t$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>c</mi> <mn>1</mn> </msub> <mi>t</mi> <mo>≤</mo> <mi>x</mi> <mo>≤</mo> <msub> <mi>c</mi> <mn>2</mn> </msub> <mi>t</mi> </mrow> </math></EquationSource> </InlineEquation>, and a semi-wave for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$c_2t\le x\le h(t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>c</mi> <mn>2</mn> </msub> <mi>t</mi> <mo>≤</mo> <mi>x</mi> <mo>≤</mo> <mi>h</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with some <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$c_1,c_2>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>c</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>c</mi> <mn>2</mn> </msub> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

Borderline behavior and propagating terrace for a free boundary problem with positive bistable nonlinearity under Dirichlet boundary conditions

期刊论文
发表日期：

2025年5月13日

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Yuki Kaneko,
Yoshio Yamada

摘要

We continue to study a free boundary problem $u_t=du_{xx}+f(u)$ $u_{t} = d u_{xx} + f (u)$ for $t>0$ $t > 0$ and $0<x<h(t)$ $0 < x < h (t)$ with positive bistable nonlinearity f under Dirichlet boundary conditions, for which Endo et al. (Discrete Contin Dyn Syst 40:3375–3394, 2020) classified large-time behaviors of solutions into three groups: the big spreading, small spreading and vanishing. Our first aim is to classify the small spreading solutions into two types; one is a standard small spreading solution and the other is a spreading solution with a moving one-peak. The latter type of spreading solution will be understood to play a borderline-behavior between the big and small spreading. When we discuss large-time behaviors of spreading solutions, it is known that a solution of the corresponding semi-wave problem allows us to obtain sharp asymptotic profiles and estimates for spreading solutions. However, the semi-wave problem for a big spreading solution has no solutions under certain circumstances. Our second aim is to derive sharp asymptotic profiles and estimates for any big spreading solution even in this situation. We will show that such a big spreading solution converges as $t\rightarrow \infty $ $t \to \infty$ to a so called propagating terrace which consists of a stationary solution for $0\le x\le c_1t$ $0 \leq x \leq c_{1} t$ , a travelling wave for $c_1t\le x\le c_2t$ $c_{1} t \leq x \leq c_{2} t$ , and a semi-wave for $c_2t\le x\le h(t)$ $c_{2} t \leq x \leq h (t)$ with some $c_1,c_2>0$ $c_{1}, c_{2} > 0$ .