کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
4610753 | 1338584 | 2013 | 9 صفحه PDF | دانلود رایگان |
![عکس صفحه اول مقاله: Non-existence of local solutions for semilinear heat equations of Osgood type Non-existence of local solutions for semilinear heat equations of Osgood type](/preview/png/4610753.png)
We establish non-existence results for the Cauchy problem of some semilinear heat equations with non-negative initial data and locally Lipschitz, non-negative source term f . Global (in time) solutions of the scalar ODE v˙=f(v) exist for v(0)>0v(0)>0 if and only if the Osgood-type condition ∫1∞dsf(s)=∞ holds; by comparison this ensures the existence of global classical solutions of ut=Δu+f(u)ut=Δu+f(u) for bounded initial data u0∈L∞(Rn)u0∈L∞(Rn). It is natural to ask whether the Osgood condition is sufficient to ensure that the problem still admits global solutions if the initial data is in Lq(Rn)Lq(Rn) for some 1⩽q<∞1⩽q<∞. Here we answer this question in the negative, and in fact show that there are initial conditions for which there exists no local solution in Lloc1(Rn) for t>0t>0.
Journal: Journal of Differential Equations - Volume 255, Issue 10, 15 November 2013, Pages 3020–3028