Boundedness of Global Solutions for a Heat Equation with Exponential Gradient Source

and Applied Analysis 3 For the case A > Ac, we improve the result by removing the restrictions u0 ≥ 0 and u0 x ≥ 0 on the initial data. Then all solutions of 1.1 blow up in finite time in C1 norm. Remark 1.2. In the critical case A Ac, all solutions have to blow up in C1 in either finite or infinite time. Moreover, if 3 occurs, then the solution will converge in C 0, 1 to the singular steady-state VAc , as t → ∞. This follows from Proposition 3.2 below. However, the possibility of 3 remains an open problem in this case. We conjecture that this could occur. As a consequence of our results, we exhibit the following interesting situation: although C1 boundedness of global solutions is true, the global solutions of 1.1 do not satisfy a uniform a priori estimate, that is, the supremum in 1 cannot be estimated in terms of the norm of the initial data. In other words, there exists a bounded, even compact, subset S ⊂ X, such that the trajectories starting from S describe an unbounded subset of X, although each of them is individually bounded and converges to the same limit. As a further consequence, the existence time T ∗, defined as a function from X into 0,∞ , is not upper semi continuous. Proposition 1.3. Assume 0 < A < Ac. There exists u0 ∈ X and a sequence {u0,n} in X with the following properties: a u0,n → u0 in C1, b T ∗ u0,n ∞ for each n, and T ∗ u0 < ∞, c supt≥0‖ un x ·, t ‖∞ : Kn → ∞. To explain the ideas of our proof, let us first recall that, in a classical paper 3 , Zelenyak showed that any one-dimensional quasilinear uniformly parabolic equation possesses a strict Lyapunov’s functional, of the form: