<article-title xmlns:mml="http://www.w3.org/1998/Math/MathML">Global Existence of Solutions to a Class of Reaction–Diffusion Systems on <inline-formula id="I1"><mml:math id="m1"><mml:msup><mml:mi mathvariant="double-struck">R</mml:mi><mml:mi>n</mml:mi></m
<p id="p1">We prove in this work the existence of a unique global nonnegative classical solution to the class of reaction–diffusion systems<div id="df_" class="formula panel"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m2"><mml:mtable columnalign="center" displaystyle="true"><...
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Capítulo de libro |
| Lenguaje: | inglés |
| Publicado: |
IntechOpen
2021
|
| Materias: | |
| Acceso en línea: | https://www.intechopen.com/chapters/75388 |
| Etiquetas: |
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| Sumario: | <p id="p1">We prove in this work the existence of a unique global nonnegative classical solution to the class of reaction–diffusion systems<div id="df_" class="formula panel"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m2"><mml:mtable columnalign="center" displaystyle="true"><mml:mtr><mml:mtd><mml:msub><mml:mi>u</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mfenced open="(" close=")" separators=","><mml:mi>t</mml:mi><mml:mi>x</mml:mi></mml:mfenced><mml:mo>=</mml:mo><mml:mi>a</mml:mi><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>u</mml:mi><mml:mfenced open="(" close=")" separators=","><mml:mi>t</mml:mi><mml:mi>x</mml:mi></mml:mfenced><mml:mo>−</mml:mo><mml:mi>g</mml:mi><mml:mfenced open="(" close=")"><mml:mi>u</mml:mi></mml:mfenced><mml:msup><mml:mi>v</mml:mi><mml:mi>m</mml:mi></mml:msup><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mi>v</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mfenced open="(" close=")" separators=","><mml:mi>t</mml:mi><mml:mi>x</mml:mi></mml:mfenced><mml:mo>=</mml:mo><mml:mi>d</mml:mi><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>v</mml:mi><mml:mfenced open="(" close=")" separators=","><mml:mi>t</mml:mi><mml:mi>x</mml:mi></mml:mfenced><mml:mo>+</mml:mo><mml:mi>λ</mml:mi><mml:mfenced open="(" close=")" separators=","><mml:mi>t</mml:mi><mml:mi>x</mml:mi></mml:mfenced><mml:mi>g</mml:mi><mml:mfenced open="(" close=")"><mml:mi>u</mml:mi></mml:mfenced><mml:msup><mml:mi>v</mml:mi><mml:mi>m</mml:mi></mml:msup><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><span class="equ"></span></div></p><p id="p8000">where <span class="inline-formula"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m3"><mml:mspace width="0.5em"></mml:mspace><mml:mi>a</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:math></span>, <span class="inline-formula"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m4"><mml:mi>d</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:math></span>, <span class="inline-formula"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m5"><mml:mi>t</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:math></span><span class="inline-formula"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m6"><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:msup><mml:mi mathvariant="double-struck">R</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:math></span>, <span class="inline-formula"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m7"><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>m</mml:mi><mml:mo>∈</mml:mo><mml:msup><mml:mi mathvariant="double-struck">N</mml:mi><mml:mo>∗</mml:mo></mml:msup></mml:math></span>, <span class="inline-formula"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m8"><mml:mi>λ</mml:mi></mml:math></span> is a nonnegative bounded function with <span class="inline-formula"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m9"><mml:mi>λ</mml:mi><mml:mfenced open="(" close=")" separators=","><mml:mi>t</mml:mi><mml:mo>.</mml:mo></mml:mfenced><mml:mo>∈</mml:mo><mml:mi mathvariant="italic">BUC</mml:mi><mml:mfenced open="(" close=")"><mml:msup><mml:mi mathvariant="double-struck">R</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:mfenced></mml:math></span> for all <span class="inline-formula"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m10"><mml:mi>t</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi mathvariant="double-struck">R</mml:mi><mml:mo>+</mml:mo></mml:msub></mml:math></span>, the initial data <span class="inline-formula"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m11"><mml:msub><mml:mi>u</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></span>, <span class="inline-formula"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m12"><mml:msub><mml:mi>v</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></span><span class="inline-formula"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m13"><mml:mo>∈</mml:mo><mml:mi mathvariant="italic">BUC</mml:mi><mml:mfenced open="(" close=")"><mml:msup><mml:mi mathvariant="double-struck">R</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:mfenced></mml:math></span>, <span class="inline-formula"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m14"><mml:mi>g</mml:mi></mml:math></span><span class="inline-formula"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m15"><mml:mo>:</mml:mo><mml:mi mathvariant="italic">BUC</mml:mi><mml:mfenced open="(" close=")"><mml:msup><mml:mi mathvariant="double-struck">R</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:mfenced><mml:mo>→</mml:mo><mml:mi mathvariant="italic">BUC</mml:mi><mml:mfenced open="(" close=")"><mml:msup><mml:mi mathvariant="double-struck">R</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:mfenced></mml:math></span> is a of class <span class="inline-formula"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m16"><mml:msup><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msup><mml:mo>,</mml:mo></mml:math></span><span class="inline-formula"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m17"><mml:mfrac><mml:mrow><mml:mi mathvariant="italic">dg</mml:mi><mml:mfenced open="(" close=")"><mml:mi>u</mml:mi></mml:mfenced></mml:mrow><mml:mi mathvariant="italic">du</mml:mi></mml:mfrac><mml:mspace width="0.5em"></mml:mspace><mml:mo>∈</mml:mo><mml:msup><mml:mi>L</mml:mi><mml:mo>∞</mml:mo></mml:msup><mml:mfenced open="(" close=")"><mml:mi mathvariant="double-struck">R</mml:mi></mml:mfenced></mml:math></span>, <span class="inline-formula"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m18"><mml:mi>g</mml:mi><mml:mfenced open="(" close=")"><mml:mn>0</mml:mn></mml:mfenced><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></span> and <span class="inline-formula"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m19"><mml:mi>g</mml:mi><mml:mfenced open="(" close=")"><mml:mi>u</mml:mi></mml:mfenced><mml:mo>≥</mml:mo><mml:mn>0</mml:mn></mml:math></span> for all <span class="inline-formula"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m20"><mml:mi>u</mml:mi><mml:mo>≥</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo></mml:math></span> The ideas of the proof is inspired from the work of Collet and Xin who proved the same result in the particular case <span class="inline-formula"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m21"><mml:mi>d</mml:mi><mml:mo>></mml:mo><mml:mi>a</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></span>, <span class="inline-formula"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m22"><mml:mi>λ</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo></mml:math></span><span class="inline-formula"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m23"><mml:mi>g</mml:mi><mml:mfenced open="(" close=")"><mml:mi>u</mml:mi></mml:mfenced><mml:mo>=</mml:mo><mml:mi>u</mml:mi><mml:mo>.</mml:mo></mml:math></span> Moreover, they showed that the <span class="inline-formula"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m24"><mml:msup><mml:mi>L</mml:mi><mml:mo>∞</mml:mo></mml:msup></mml:math></span>-norm of <span class="inline-formula"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m25"><mml:mi>v</mml:mi></mml:math></span> can not grow faster than <span class="inline-formula"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m26"><mml:mi>O</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mo>ln</mml:mo><mml:mo>ln</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mfenced></mml:math></span> for any space dimension.</p> |
|---|---|
| ISBN: | 978-1-83968-656-6 |
| DOI: | 10.5772/intechopen.95543 |