<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$d \in \{3,4,5,\ldots \}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>∈</mo> <mo stretchy="false">{</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mo>…</mo> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Omega \subset \mathbb {R}^d$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> be open bounded with Lipschitz boundary. Let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$Q = \Omega \times (0,\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Q</mi> <mo>=</mo> <mi mathvariant="normal">Ω</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$p \in C({\overline{Q}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mover> <mi>Q</mi> <mo>¯</mo> </mover> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be such that <Equation ID="Equ40"> <EquationSource Format="TEX">$ 2< p^- \le p(\cdot ) \le p^+ < \frac{2^}{2}+1, $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mn>2</mn> <mo><</mo> <msup> <mi>p</mi> <mo>-</mo> </msup> <mo>≤</mo> <mi>p</mi> <mrow> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <msup> <mi>p</mi> <mo>+</mo> </msup> <mo><</mo> <mfrac> <msup> <mn>2</mn> <mo>∗</mo> </msup> <mn>2</mn> </mfrac> <mo>+</mo> <mn>1</mn> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$2^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>2</mn> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation> is the critical Sobolev exponent of 2, <Equation ID="Equ41"> <EquationSource Format="TEX">$ p^- := \mathop {\mathrm {ess\,inf}}\limits _{(x,t) \in Q} p(x,t) \quad \text {and}\quad p^+ := \mathop {\mathrm {ess\,sup}}\limits _{(x,t) \in Q} p(x,t). $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msup> <mi>p</mi> <mo>-</mo> </msup> <mo>:</mo> <mo>=</mo> <munder> <mrow> <mi mathvariant="normal">ess</mi> <mspace width="0.166667em" /> <mi mathvariant="normal">inf</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>Q</mi> </mrow> </munder> <mi>p</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="1em" /> <mtext>and</mtext> <mspace width="1em" /> <msup> <mi>p</mi> <mo>+</mo> </msup> <mo>:</mo> <mo>=</mo> <munder> <mrow> <mi mathvariant="normal">ess</mi> <mspace width="0.166667em" /> <mi mathvariant="normal">sup</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>Q</mi> </mrow> </munder> <mi>p</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </Equation>Consider the reaction-diffusion parabolic problem <Equation ID="Equ42"> <EquationSource Format="TEX">$ (P) \quad \left\{ \begin{array}{ll} \displaystyle \frac{u_t}{|x|^2} - \Delta u = k(t) \, |u|^{p(x,t)-2}u & (x,t) \in \Omega \times (0,T), \\ u(x,t) = 0, & (x,t) \in \partial \Omega \times (0,T), \\ u(x,0) = u_0(x), & x \in \Omega , \end{array}\right. $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="1em" /> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mfrac> <msub> <mi>u</mi> <mi>t</mi> </msub> <msup> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> </mfrac> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>=</mo> <mi>k</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="0.166667em" /> <msup> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> </mrow> </mstyle> </mtd> <mtd columnalign="left"> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>u</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq6"> <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="IEq7"> <EquationSource Format="TEX">$0 \ne u_0 \in W^{1,2}_0(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≠</mo> <msub> <mi>u</mi> <mn>0</mn> </msub> <mo>∈</mo> <msubsup> <mi>W</mi> <mn>0</mn> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We investigate the existence and uniqueness of a weak solution to (<i>P</i>). The upper and lower bounds on the blow-up time of the weak solution are also considered.</p>

Existence, uniqueness and blow-up estimates for a reaction-diffusion equation with p(x, t)-exponent in the forcing term

期刊论文
发表日期：

2025年12月10日

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Nguyen Thanh Tung,
Le Xuan Truong,
Tan Duc Do

摘要

Let $d \in \{3,4,5,\ldots \}$ $d \in {3, 4, 5, \dots}$ and $\Omega \subset \mathbb {R}^d$ $Ω \subset R^{d}$ be open bounded with Lipschitz boundary. Let $Q = \Omega \times (0,\infty )$ $Q = Ω \times (0, \infty)$ and $p \in C({\overline{Q}})$ $p \in C (\bar{Q})$ be such that $ 2< p^- \le p(\cdot ) \le p^+ < \frac{2^*}{2}+1, $ $2 < p^{-} \leq p (\cdot) \leq p^{+} < \frac{2^{*}}{2} + 1,$ where $2^*$ $2^{*}$ is the critical Sobolev exponent of 2, $ p^- := \mathop {\mathrm {ess\,inf}}\limits _{(x,t) \in Q} p(x,t) \quad \text {and}\quad p^+ := \mathop {\mathrm {ess\,sup}}\limits _{(x,t) \in Q} p(x,t). $ $p^{-} : = \underset{(x, t) \in Q}{ess \inf} p (x, t) and p^{+} : = \underset{(x, t) \in Q}{ess \sup} p (x, t) .$ Consider the reaction-diffusion parabolic problem $ (P) \quad \left\{ \begin{array}{ll} \displaystyle \frac{u_t}{|x|^2} - \Delta u = k(t) \, |u|^{p(x,t)-2}u & (x,t) \in \Omega \times (0,T), \\ u(x,t) = 0, & (x,t) \in \partial \Omega \times (0,T), \\ u(x,0) = u_0(x), & x \in \Omega , \end{array}\right. $ $(P) \{\begin{matrix} \frac{u_{t}}{{| x |}^{2}} - Δ u = k (t) {| u |}^{p (x, t) - 2} u & (x, t) \in Ω \times (0, T), \\ u (x, t) = 0, & (x, t) \in \partial Ω \times (0, T), \\ u (x, 0) = u_{0} (x), & x \in Ω, \end{matrix})$ where $T > 0$ $T > 0$ and $0 \ne u_0 \in W^{1,2}_0(\Omega )$ $0 \neq u_{0} \in W_{0}^{1, 2} (Ω)$ . We investigate the existence and uniqueness of a weak solution to (P). The upper and lower bounds on the blow-up time of the weak solution are also considered.