Self-Similar Blow-Up in Higher-Order Semilinear Parabolic Equations
Self-Similar Blow-Up in Higher-Order Semilinear Parabolic Equations
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DOI：
10.1137/s003613990241552x
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发表时间：
2004
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影响因子：
--
通讯作者：
C. Budd;J. F. Williams;V. Galaktionov
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文献类型：
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作者：
C. Budd;J. F. Williams;V. Galaktionov
本文研究了在$\re \times \re_+$中对于具有有界初始数据u0 (x)的形式为（$D_x=\partial/\partial x$） \[ u_t = \Dx u + |u|^{p-1}u, \,\,\,\mbox{where} \,\, \,\,\,p > 1, \quad \mbox{ and } \quad u_t = \Dx u + e^u \]的一维2m阶，m>1的半线性抛物型偏微分方程的柯西问题。具体地说，我们对那些在有限时间t内在原点爆炸的解感兴趣。我们证明，与燃烧理论中的经典二阶抛物方程ut = uxx + up和ut = uxx + eu的解相比，它们的高阶对应方程的爆炸是渐近自相似的。特别地，存在精确的非平凡自相似爆破解，对于多项式非线性，u* (x,t) = (t -t)-1/(p-1)f (y)；对于指数非线性，u(x,t) = -ln(t -t) + f(y)，其中y= x/(t -t)1/2m为后向高阶热核变量。曲线f(y)满足具有相同非自伴随高阶线性差分的相关半线性ode。
We study the Cauchy problem in $\re \times \re_+$ for one-dimensional 2mth-order, m>1, semilinear parabolic PDEs of the form ($D_x=\partial/\partial x$) \[ u_t = \Dx u + |u|^{p-1}u, \,\,\,\mbox{where} \,\, \,\,\,p > 1, \quad \mbox{ and } \quad u_t = \Dx u + e^u \] with bounded initial data u0 (x). Specifically, we are interested in those solutions that blow up at the origin in a finite time T. We show that, in contrast to the solutions of the classical second-order parabolic equations ut = uxx + up and ut = uxx + eu from combustion theory, the blow-up in their higher-order counterparts is asymptotically self-similar. In particular, there exist exact nontrivial self-similar blow-up solutions, u* (x,t) = (T-t)-1/(p-1)f (y) in the case of the polynomial nonlinearity and u(x,t) = -ln(T-t) + f(y) for the exponential nonlinearity, where y= x/(T-t)1/2m is the backward higher-order heat kernel variable. The profiles f(y) satisfy related semilinear ODEs that share the same non--self-adjoint higher-order linear dif...