Volume 28, Issue 2
Blowup, Global Fast and Slow Solutions for a Semilinear Combustible System

J. Part. Diff. Eq., 28 (2015), pp. 139-157.

Published online: 2015-06

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• Abstract

In this paper, we investigate a semilinear combustible system $u_t-du_{xx}=v^p, v_t-dv_{xx}=u^q$ with double fronts free boundary, where p ≥ 1, q ≥ 1. For such a problem, we use the contraction mapping theorem to prove the local existence and uniqueness of the solution. Also we study the blowup and global existence property of the solution. Our results show that when pq › 1 blowup occurs if the initial datum is large enough and the solution is global and slow, whose decay rate is at most polynomial if the initial value is suitably large, while when p › 1, q › 1 there is a global and fast solution, which decays uniformly at an exponential rate if the initial datum is small.

• Keywords

Free boundary blowup global fast solution global slow solution

• AMS Subject Headings

35K20, 35R35, 92B05

@Article{JPDE-28-139, author = {Yuan , Junli }, title = {Blowup, Global Fast and Slow Solutions for a Semilinear Combustible System}, journal = {Journal of Partial Differential Equations}, year = {2015}, volume = {28}, number = {2}, pages = {139--157}, abstract = { In this paper, we investigate a semilinear combustible system $u_t-du_{xx}=v^p, v_t-dv_{xx}=u^q$ with double fronts free boundary, where p ≥ 1, q ≥ 1. For such a problem, we use the contraction mapping theorem to prove the local existence and uniqueness of the solution. Also we study the blowup and global existence property of the solution. Our results show that when pq › 1 blowup occurs if the initial datum is large enough and the solution is global and slow, whose decay rate is at most polynomial if the initial value is suitably large, while when p › 1, q › 1 there is a global and fast solution, which decays uniformly at an exponential rate if the initial datum is small.}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v28.n2.4}, url = {http://global-sci.org/intro/article_detail/jpde/5107.html} }
TY - JOUR T1 - Blowup, Global Fast and Slow Solutions for a Semilinear Combustible System AU - Yuan , Junli JO - Journal of Partial Differential Equations VL - 2 SP - 139 EP - 157 PY - 2015 DA - 2015/06 SN - 28 DO - http://doi.org/10.4208/jpde.v28.n2.4 UR - https://global-sci.org/intro/article_detail/jpde/5107.html KW - Free boundary KW - blowup KW - global fast solution KW - global slow solution AB - In this paper, we investigate a semilinear combustible system $u_t-du_{xx}=v^p, v_t-dv_{xx}=u^q$ with double fronts free boundary, where p ≥ 1, q ≥ 1. For such a problem, we use the contraction mapping theorem to prove the local existence and uniqueness of the solution. Also we study the blowup and global existence property of the solution. Our results show that when pq › 1 blowup occurs if the initial datum is large enough and the solution is global and slow, whose decay rate is at most polynomial if the initial value is suitably large, while when p › 1, q › 1 there is a global and fast solution, which decays uniformly at an exponential rate if the initial datum is small.