具渗透或蒸发项一维渗流方程的源型解

对具对流项渗流方程ul=(um)xx+(un)x,(x,t)∈R×(0,T)的Cauchy问题进行研究,其中m>1,n>0,当初值为Dirac测度时的解(通常称之为源型解Source-typesolution)。方程在{u=0}处蜕化;带对流项(un)x,故方程具有双曲性;当n<1时,方程的对流项在{u=0}上是奇异的;方程不具吸附项。方程的上述这些特征,给源型解的研究造成实质性的困难,以往文献中对标准渗流方程或具有吸附项渗流方程的研究方法不能推广到此类方程。但在应用了Moser迭代、微分不等式、Bernstein方法、积分估计等一系列技巧后,克服了难点,建立了源型解存在性、唯一性、正则性和渐近性等相关结果。

Source-type Solutions of Porous Medium Equations with Infiltration or Evaporation in One-Dimension

In this paperwe considerthe equationu l =(u m ) xx +(u n ) x ,(x,t)∈R×(0,T) with a measure as its initial conditions.Whenthe measureis bounded,the theories of existence,uniqueness and theasymptotic behavior near(0,0)for the source-typesolution of the equationhave been established.The resultsimprove the previous works by the author.Herewe get rid of the certainhypothesesof the parameters m and n in the previousresults.