The quasilinear parabolic equation with an absorption potential is considered:|u|q?1u t? ?p(u) = ?b(t, x)|u|λ?1u (t, x) ∈ (0, T) × ?, λ p q 0,where ? is a bounded smooth domain in Rn, n 1, b is an absorptionpotential which is a continuous function such that b(t, x) 0 in [0, T) ×? and b(t, x) ≡ 0 in {T} × ?. In the paper, the conditions for b(t, x) thatguarantee the uniform boundedness of an arbitrary weak solution of thementioned equation in an arbitrary subdomain ?0 : ?0 ? ? are considered.Under the above conditions the sharp upper estimate for all weak solutionsu is obtained. The estimate holds for the solutions of the equation witharbitrary initial and boundary data, including blow-up data (provided thatsuch a solution exists), namely, u = ∞ on {0} × ?, u = ∞ on (0, T) × ??.
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机译：考虑具有吸收电势的拟线性抛物方程：| u | q？1u t？ ？p（u）=？b（t，x）| u |λ？1u（t，x）∈（0，T）×？，λ> p> q> 0，其中？是Rn中的有界光滑域，n> 1，b是一个吸收势，它是一个连续函数，使得[0，T）×b中的b（t，x）> 0。 {T}×？中的b（t，x）≡0。在本文中，b（t，x）的条件保证了在任意子域中所述方程的任意弱解的一致有界θ0：θ0？ ？在上述条件下，可以得到所有弱解的尖锐上估计。该估计适用于具有任意初始和边界数据的方程的解，包括爆破数据（假设存在这样的解），即，{0}×？上的u =∞，（0，T）×上的u =∞ ??。
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