### A Classification of Solutions of Certain Nonlinear Differential Inequalities with Application to Theorems of Liouville Type.

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We study the absence of nonnegative global solutions to parabolic inequalities of the type ${u}_{t}\ge -{(-\Delta )}^{\beta /2}u-V\left(x\right)u+h(x,t){u}^{p}$, where ${(-\Delta )}^{\beta /2}$, 0 < β ≤ 2, is the β/2 fractional power of the Laplacian. We give a sufficient condition which implies that the only global solution is trivial if p > 1 is small. Among other properties, we derive a necessary condition for the existence of local and global nonnegative solutions to the above problem for the function V satisfying ${V\u208a\left(x\right)\sim a\left|x\right|}^{-b}$, where a ≥ 0, b > 0, p > 1 and V₊(x): = maxV(x),0. We...

We investigate critical exponents for blow-up of nonnegative solutions to a class of parabolic inequalities. The proofs make use of a priori estimates of solutions combined with a simple scaling argument.

We develop a new method for proving the existence of a boundary trace, in the class of Borel measures, of nonnegative solutions of $-\Delta u+g(x,u)\ge 0$ in a smooth domain $\Omega $ under very general assumptions on $g$. This new definition which extends the previous notions of boundary trace is based upon a sweeping technique by solutions of Dirichlet problems with measure boundary data. We also prove a boundary pointwise blow-up estimate of any solution of such inequalities in terms of the Poisson kernel. If the nonlinearity...

We consider the Darboux problem for a functional differential equation: $\left(\partial \xb2u\right)/\left(\partial x\partial y\right)(x,y)=f(x,y,{u}_{(x,y)},u(x,y),\partial u/\partial x(x,y),\partial u/\partial y(x,y))$ a.e. in [0,a]×[0,b], u(x,y) = ψ(x,y) on [-a₀,a]×[-b₀,b]∖(0,a]×(0,b], where the function ${u}_{(x,y)}:[-a\u2080,0]\times [-b\u2080,0]\to {\mathbb{R}}^{k}$ is defined by ${u}_{(x,y)}(s,t)=u(s+x,t+y)$ for (s,t) ∈ [-a₀,0]×[-b₀,0]. We give a few theorems about weak and strong inequalities for this problem. We also discuss the case where the right-hand side of the differential equation is linear.

The paper deals with a weakly coupled system of functional-differential equations ${\partial}_{t}{u}_{i}(t,x)={f}_{i}(t,x,u(t,x),u,{\partial}_{x}{u}_{i}(t,x),{\partial}_{xx}{u}_{i}(t,x))$, i ∈ S, where (t,x) = (t,x₁,...,xₙ) ∈ (0,a) × G, $u={{u}_{i}}_{i\in S}$ and S is an arbitrary set of indices. Initial boundary conditions are considered and the following questions are discussed: estimates of solutions, criteria of uniqueness, continuous dependence of solutions on given functions. The right hand sides of the equations satisfy nonlinear estimates of the Perron type with respect to the unknown functions. The results are...

We give a unified statement and proof of a class of well known mean value inequalities for nonnegative functions with a nonlinear bound on the Laplacian. We generalize these to domains with boundary, requiring a (possibly nonlinear) bound on the normal derivative at the boundary. These inequalities give rise to an energy quantization principle for sequences of solutions of boundary value problems that have bounded energy and whose energy densities satisfy nonlinear bounds on the Laplacian and normal...