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• W2975675791 abstract "We study the global existence of solutions to the Cauchy problem for the wave equation with time-dependent damping and a power nonlinearity in the overdamping case: <disp-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartLayout 1st Row StartLayout Enlarged left-brace 1st Row 1st Column u Subscript t t Baseline minus normal upper Delta u plus b left-parenthesis t right-parenthesis u Subscript t Baseline equals upper N left-parenthesis u right-parenthesis comma 2nd Column a m p semicolon t element-of left-bracket 0 comma upper T right-parenthesis comma x element-of double-struck upper R Superscript d Baseline comma 2nd Row 1st Column u left-parenthesis 0 right-parenthesis equals u 0 comma u Subscript t Baseline left-parenthesis 0 right-parenthesis equals u 1 comma 2nd Column a m p semicolon x element-of double-struck upper R Superscript d Baseline period EndLayout EndLayout> <mml:semantics> <mml:mtable columnalign=right left right left right left right left right left right left rowspacing=3pt columnspacing=0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em side=left displaystyle=true> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mo>{</mml:mo> <mml:mtable columnalign=left left rowspacing=4pt columnspacing=1em> <mml:mtr> <mml:mtd> <mml:msub> <mml:mi>u</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>t</mml:mi> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> <mml:mo>−<!-- − --></mml:mo> <mml:mi mathvariant=normal>Δ<!-- Δ --></mml:mi> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>,</mml:mo> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mi>t</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mo stretchy=false>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>,</mml:mo> <mml:mtext> </mml:mtext> <mml:mi>x</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mi>u</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mtext> </mml:mtext> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mi>x</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo>.</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> <mml:mo fence=true stretchy=true symmetric=true /> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> <mml:annotation encoding=application/x-tex>begin{align*}left {begin {array}{ll} u_{tt} - Delta u + b(t) u_t = N(u),&tin [0,T), xin mathbb {R}^d, u(0) = u_0, u_t(0) = u_1,&xin mathbb {R}^d. end{array}right . end{align*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> Here, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=b left-parenthesis t right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>b(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a positive <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C Superscript 1> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding=application/x-tex>C^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-function on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-bracket 0 comma normal infinity right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>[0,infty )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfying <disp-formula content-type=math/mathml> [ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=b left-parenthesis t right-parenthesis Superscript negative 1 Baseline element-of upper L Superscript 1 Baseline left-parenthesis 0 comma normal infinity right-parenthesis comma> <mml:semantics> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>b(t)^{-1} in L^1(0,infty ),</mml:annotation> </mml:semantics> </mml:math> ] </disp-formula>whose case is called <italic>overdamping</italic>. <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper N left-parenthesis u right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>N(u)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denotes the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=application/x-tex>p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>th order power nonlinearities. It is well known that the problem is locally well-posed in the energy space <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H Superscript 1 Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis times upper L squared left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mo>×<!-- × --></mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>H^1(mathbb {R}^d)times L^2(mathbb {R}^d)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the energy-subcritical or energy-critical case <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=1 less-than-or-equal-to p less-than-or-equal-to p 1> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>p</mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:msub> <mml:mi>p</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>1le ple p_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p 1 colon equals 1 plus StartFraction 4 Over d minus 2 EndFraction> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>p</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>:=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mfrac> <mml:mn>4</mml:mn> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:mfrac> </mml:mrow> <mml:annotation encoding=application/x-tex>p_1:=1+frac {4}{d-2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d greater-than-or-equal-to 3> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>dge 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p 1 equals normal infinity> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>p</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>p_1=infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d equals 1 comma 2> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>d=1,2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It is known that when <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper N left-parenthesis u right-parenthesis colon equals plus-or-minus StartAbsoluteValue u EndAbsoluteValue Superscript p> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>:=</mml:mo> <mml:mo>±<!-- ± --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mi>u</mml:mi> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>N(u):=pm |u|^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, small data blow-up in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L Superscript 1> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding=application/x-tex>L^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-framework occurs in the case <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=b left-parenthesis t right-parenthesis Superscript negative 1 not-an-element-of upper L Superscript 1 Baseline left-parenthesis 0 comma normal infinity right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>∉<!-- ∉ --></mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>b(t)^{-1} notin L^1(0,infty )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=1 greater-than p greater-than p Subscript c Baseline left-parenthesis greater-than p 1 right-parenthesis> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>></mml:mo> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mo>></mml:mo> <mml:msub> <mml:mi>p</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>1>p>p_c(> p_1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p Subscript c> <mml:semantics> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>p_c</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a critical exponent, i.e., threshold exponent dividing the small data global existence and the small data blow-up. The main purpose in the present paper is to prove the global well-posedness to the problem for small data <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis u 0 comma u 1 right-parenthesis element-of upper H Superscript 1 Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis times upper L squared left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy=false>)</mml:mo> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mo>×<!-- × --></mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(u_0,u_1)in H^1(mathbb {R}^d)times L^2(mathbb {R}^d)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the whole energy-subcritical case, i.e., <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=1 less-than-or-equal-to p greater-than p 1> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:msub> <mml:mi>p</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>1le p>p_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This result implies that the small data blow-up does not occur in the overdamping case, different from the other case <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=b left-parenthesis t right-parenthesis Superscript negative 1 not-an-element-of upper L Superscript 1 Baseline left-parenthesis 0 comma normal infinity right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>∉<!-- ∉ --></mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>b(t)^{-1}notin L^1(0,infty )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, i.e., the effective or noneffective damping." @default.
• W2975675791 created "2019-10-03" @default.
• W2975675791 creator A5026324990 @default.
• W2975675791 creator A5044169685 @default.
• W2975675791 date "2019-09-25" @default.
• W2975675791 modified "2023-09-26" @default.
• W2975675791 title "Global well-posedness for the semilinear wave equation with time dependent damping in the overdamping case" @default.
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• W2975675791 doi "https://doi.org/10.1090/proc/14297" @default.
• W2975675791 hasPublicationYear "2019" @default.
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