On the asymptotic behavior of solutions of second order parabolic partial differential equations

Wei-Cheng Lian; Cheh-Chih Yeh

Annales Polonici Mathematici (1996)

  • Volume: 63, Issue: 3, page 223-234
  • ISSN: 0066-2216

Abstract

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We consider the second order parabolic partial differential equation    ∑ i , j = 1 n a i j ( x , t ) u x i x j + ∑ i = 1 n b i ( x , t ) u x i + c ( x , t ) u - u t = 0 . Sufficient conditions are given under which every solution of the above equation must decay or tend to infinity as |x|→ ∞. A sufficient condition is also given under which every solution of a system of the form    L α [ u α ] + ∑ β = 1 N c α β ( x , t ) u β = f α ( x , t ) , where    L α [ u ] ≡ ∑ i , j = 1 n a i j α ( x , t ) u x i x j + ∑ i = 1 n b i α ( x , t ) u x i - u t , must decay as t → ∞.

How to cite

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Wei-Cheng Lian, and Cheh-Chih Yeh. "On the asymptotic behavior of solutions of second order parabolic partial differential equations." Annales Polonici Mathematici 63.3 (1996): 223-234. <http://eudml.org/doc/262828>.

@article{Wei1996,
abstract = {We consider the second order parabolic partial differential equation    $∑^n_\{i,j=1\} a_\{ij\}(x,t) u_\{x_\{i\}x_\{j\}\} + ∑^n_\{i=1\} b_i(x,t) u_\{x_i\} + c(x,t)u - u_t = 0$. Sufficient conditions are given under which every solution of the above equation must decay or tend to infinity as |x|→ ∞. A sufficient condition is also given under which every solution of a system of the form    $L^α[u^α] + ∑_\{β=1\}^N c^\{αβ\}(x,t) u^β = f^α(x,t)$, where    $L^α[u] ≡ ∑^n_\{i,j=1\} a_\{ij\}^α(x,t) u_\{x_\{i\}x_\{j\}\} + ∑^n_\{i=1\} b_i^α(x,t) u_\{x_i\} - u_t$, must decay as t → ∞.},
author = {Wei-Cheng Lian, Cheh-Chih Yeh},
journal = {Annales Polonici Mathematici},
keywords = {asymptotic behavior; second order partial differential equation; maximum principles; large time behaviour},
language = {eng},
number = {3},
pages = {223-234},
title = {On the asymptotic behavior of solutions of second order parabolic partial differential equations},
url = {http://eudml.org/doc/262828},
volume = {63},
year = {1996},
}

TY - JOUR
AU - Wei-Cheng Lian
AU - Cheh-Chih Yeh
TI - On the asymptotic behavior of solutions of second order parabolic partial differential equations
JO - Annales Polonici Mathematici
PY - 1996
VL - 63
IS - 3
SP - 223
EP - 234
AB - We consider the second order parabolic partial differential equation    $∑^n_{i,j=1} a_{ij}(x,t) u_{x_{i}x_{j}} + ∑^n_{i=1} b_i(x,t) u_{x_i} + c(x,t)u - u_t = 0$. Sufficient conditions are given under which every solution of the above equation must decay or tend to infinity as |x|→ ∞. A sufficient condition is also given under which every solution of a system of the form    $L^α[u^α] + ∑_{β=1}^N c^{αβ}(x,t) u^β = f^α(x,t)$, where    $L^α[u] ≡ ∑^n_{i,j=1} a_{ij}^α(x,t) u_{x_{i}x_{j}} + ∑^n_{i=1} b_i^α(x,t) u_{x_i} - u_t$, must decay as t → ∞.
LA - eng
KW - asymptotic behavior; second order partial differential equation; maximum principles; large time behaviour
UR - http://eudml.org/doc/262828
ER -

References

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