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)>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) for bounded initial data 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) for some 1≤q0. © 2013 Elsevier Inc.
Sierz;E;ga, M., Robinson, J. C., Laister, R., Robinson, J. C., & Sierzega, M. (2013). Non-existence of local solutions for semilinear heat equations of Osgood type. Journal of Differential Equations, 255(10), 3020-3028. https://doi.org/10.1016/j.jde.2013.07.007